Tag Archives: filter

Pure Data and libpd: Integrating with Native Code for Interactive Testing

Over the past couple of years, I’ve built up a nice library of DSP code, including effects, oscillators, and utilities. One thing that always bothered me however, is how to test this code in an efficient and reliable way. The two main methods I have used in the past have their pros and cons, but ultimately didn’t satisfy me.

One is to process an effect or generate a source into a wave file that I can open with an audio editor so I can listen to the result and examine the output. This method is okay, but it is tedious and doesn’t allow for real-time adjustment of parameters or any sort of instant feedback.

For effects like filters, I can also generate a text file containing the frequency/phase response data that I can view in a plotting application. This is useful in some ways, but this is audio — I want to hear it!

Lately I’ve gotten to know Pure Data a little more, so I thought about using it for interactive testing of my DSP modules. On its own, Pure Data does not interact with code of course, but that’s where libpd comes in. This is a great library that wraps up much of Pure Data’s functionality so that you can use it right from your own code (it works with C, C++, Objective-C, Java, and others). Here is how I integrated it with my own code to set up a nice flexible testing framework; and this is just one application of using libpd and Pure Data together — the possiblities go far beyond this!

First we start with the Pure Data patches. The receiver patch is opened and maintained in code by libpd, and has two responsiblities: 1) generate a test tone that the effect is applied to, and 2) receive messages from the control patch and dispatch them to C++.

Receiver patch, opened by libpd.

Receiver patch, opened by libpd.

The control patch is opened in Pure Data and acts as the interactive patch. It has controls for setting the frequency and volume of the synthesizer tone that acts as the source, as well as controls for the filter effect that is being tested.

Control patch, opened in Pure Data, and serves as the interactive UI for testing.

Control patch, opened in Pure Data, and serves as the interactive UI for testing.

As can be seen from the patches above, they communicate to each other via the netsend/netreceive objects by opening a port on the local machine. Since I’m only sending simple data to the receiver patch, I opted to use UDP over TCP as the network protocol. (Disclaimer: my knowledge of network programming is akin to asking “what is a for loop”).

Hopefully the purpose of these two patches is clear, so we can now move on to seeing how libpd brings it all together in code. It is worth noting that libpd does not output audio to the hardware, it only processes the data. Pure Data, for example, commonly uses Portaudio to send the audio data to the sound card, but I will be using Core Audio instead. Additionally, I’m using the C++ wrapper from libpd.

An instance of PdBase is first created with the desired intput/output channels and sample rate, and a struct contains data that will need to be held on to that will become clear further on.

struct TestData
{
    AudioUnit outputUnit;
    EffectProc effectProc;

    PdBase* pd;
    Patch pdPatch;
    float* pdBuffer;
    int pdTicks;
    int pdSamplesPerBlock;

    CFRingBuffer<float> ringBuffer;
    int maxFramesPerSlice;
    int framesInReserve;
};

int main(int argc, const char * argv[])
{
    PdBase pd;
    pd.init(0, 2, 48000); // No input needed for tests.

    TestData testData;
    testData.pd = &pd;
    testData.pdPatch = pd.openPatch("receiver.pd", ".");
}

Next, we ask Core Audio for an output Audio Unit that we can use to send audio data to the sound card.

int main(int argc, const char * argv[])
{
    PdBase pd;
    pd.init(0, 2, 48000); // No input needed for tests.

    TestData testData;
    testData.pd = &pd;
    testData.pdPatch = pd.openPatch("receiver.pd", ".");

    {
        AudioComponentDescription outputcd = {0};
        outputcd.componentType = kAudioUnitType_Output;
        outputcd.componentSubType = kAudioUnitSubType_DefaultOutput;
        outputcd.componentManufacturer = kAudioUnitManufacturer_Apple;

        AudioComponent comp = AudioComponentFindNext(NULL, &outputcd);
        if (comp == NULL)
        {
            std::cerr << "Failed to find matching Audio Unit.\n";
            exit(EXIT_FAILURE);
        }

        OSStatus error;
        error = AudioComponentInstanceNew(comp, &testData.outputUnit);
        if (error != noErr)
        {
            std::cerr << "Failed to open component for Audio Unit.\n";
            exit(EXIT_FAILURE);
        }

        Float64 sampleRate = 48000;
        UInt32 dataSize = sizeof(sampleRate);
        error = AudioUnitSetProperty(audioUnit,
                                     kAudioUnitProperty_SampleRate,
                                     kAudioUnitScope_Input,
                                     0, &sampleRate, dataSize);

        AudioUnitInitialize(audioUnit);
    }
}

The next part needs some explanation, because we need to consider how the Pure Data patch interacts with Core Audio’s render callback function that we will provide. This function will be called continuously on a high priority thread with a certain number of frames that we need to fill with audio data. Pure Data, by default, processes 64 samples per channel per block. It’s unlikely that these two numbers (the number of frames that Core Audio wants and the number of frames processed by Pure Data) will always agree. For example, in my initial tests, Core Audio specified its maximum block size to be 512 frames, but it actually asked for 470 & 471 (alternating) when it ran. Rather than trying to force the two to match block sizes, I use a ring buffer as a medium between the two — that is, read sample data from the opened Pure Data patch into the ring buffer, and then read from the ring buffer into the buffers provided by Core Audio.

Fortunately, Core Audio can be queried for the maximum number of frames it will ask for, so this will determine the number of samples we read from the Pure Data patch. We can read a multiple of Pure Data’s 64-sample block by specifying a value for “ticks” in libpd, and this value will just be equal to the maximum frames from Core Audio divided by Pure Data’s block size. The actual number of samples read/processed will of course be multiplied by the number of channels (2 in this case for stereo).

The final point on this is to handle the case where the actual number of frames processed in a block is less than the maximum. Obviously it would only take a few blocks for the ring buffer’s write pointer to catch up with the read pointer and cause horrible audio artifacts. To account for this, I make the ring buffer twice as long as the number of samples required per block to give it some breathing room, and also keep track of the number of frames in reserve currently in the ring buffer at the end of each block. When this number exceeds the number of frames being processed in a block, no processing from the patch occurs, giving the ring buffer a chance to empty out its backlog of frames.

int main(int argc, const char * argv[])
{
    <snip> // As above.

    UInt32 framesPerSlice;
    UInt32 dataSize = sizeof(framesPerSlice);
    AudioUnitGetProperty(testData.outputUnit,
                         kAudioUnitProperty_MaximumFramesPerSlice,
                         kAudioUnitScope_Global,
                         0, &framesPerSlice, &dataSize);
    testData.pdTicks = framesPerSlice / pd.blockSize();
    testData.pdSamplesPerBlock = (pd.blockSize() * 2) * testData.pdTicks; // 2 channels for stereo output.
    testData.maxFramesPerSlice = framesPerSlice;

    AURenderCallbackStruct renderCallback;
    renderCallback.inputProc = AudioRenderProc;
    renderCallback.inputProcRefCon = &testData;
    AudioUnitSetProperty(testData.outputUnit,
                         kAudioUnitProperty_SetRenderCallback,
                         kAudioUnitScope_Input,
                         0, &renderCallback, sizeof(renderCallback));

    testData.pdBuffer = new float[testData.pdSamplesPerBlock];
    testData.ringBuffer.resize(testData.pdSamplesPerBlock * 2); // Twice as long as needed in order to give it some buffer room.
    testData.effectProc = EffectProcess;
}

With the output Audio Unit and Core Audio now set up, let’s look at the render callback function. It reads the audio data from the Pure Data patch if needed into the ring buffer, which in turn fills the buffer list provided by Core Audio. The buffer list is then passed on to the callback that processes the effect being tested.

OSStatus AudioRenderProc (void *inRefCon,
                          AudioUnitRenderActionFlags *ioActionFlags,
                          const AudioTimeStamp *inTimeStamp,
                          UInt32 inBusNumber,
                          UInt32 inNumberFrames,
                          AudioBufferList *ioData)
    {
        TestData *testData = (TestData *)inRefCon;

        // Don't require input, but libpd requires valid array.
        float inBuffer[0];

        // Only read from Pd patch if the sample excess is less than the number of frames being processed.
        // This effectively empties the ring buffer when it has enough samples for the current block, preventing the
        // write pointer from catching up to the read pointer.
        if (testData->framesInReserve < inNumberFrames)
        {
            testData->pd->processFloat(testData->pdTicks, inBuffer, testData->pdBuffer);
            for (int i = 0; i < testData->pdSamplesPerBlock; ++i)
            {
                testData->ringBuffer.write(testData->pdBuffer[i]);
            }
            testData->framesInReserve += (testData->maxFramesPerSlice - inNumberFrames);
        }
        else
        {
            testData->framesInReserve -= inNumberFrames;
        }

        // NOTE: Audio data from Pd patch is interleaved, whereas Core Audio buffers are non-interleaved.
        for (UInt32 frame = 0; frame < inNumberFrames; ++frame)
        {
            Float32 *data = (Float32 *)ioData->mBuffers[0].mData;
            data[frame] = testData->ringBuffer.read();
            data = (Float32 *)ioData->mBuffers[1].mData;
            data[frame] = testData->ringBuffer.read();
        }

        if (testData->effectCallback != nullptr)
        {
            testData->effectCallback(ioData, inNumberFrames);
        }

        return noErr;
    }

Finally, let’s see the callback function that processes the filter. It’s about as simple as it gets — it just processes the filter effect being tested on the audio signal that came from Pure Data.

void EffectProcess(AudioBufferList* audioData, UInt32 numberOfFrames)
{
    for (UInt32 frame = 0; frame < numberOfFrames; ++frame)
    {
        Float32 *data = (Float32 *)audioData->mBuffers[0].mData;
        data[frame] = filter.left.sample(data[frame]);
        data = (Float32 *)audioData->mBuffers[1].mData;
        data[frame] = filter.right.sample(data[frame]);
    }
}

Not quite done yet, though, since we need to subscribe the open libpd instance of Pure Data to the messages we want to receive from the control patch. The messages received will then be dispatched inside the C++ code to handle appropriate behavior.

int main(int argc, const char * argv[])
{
    <snip> // As above.

    pd.subscribe("fromPd_filterfreq");
    pd.subscribe("fromPd_filtergain");
    pd.subscribe("fromPd_filterbw");
    pd.subscribe("fromPd_filtertype");
    pd.subscribe("fromPd_quit");

    // Start audio processing.
    pd.computeAudio(true);
    AudioOutputUnitStart(testData.outputUnit);

    bool running = true;
    while (running)
    {
        while (pd.numMessages() > 0)
        {
            Message msg = pd.nextMessage();
            switch (msg.type)
            {
                case pd::PRINT:
                    std::cout << "PRINT: " << msg.symbol << "\n";
                    break;

                case pd::BANG:
                    std::cout << "BANG: " << msg.dest << "\n";
                    if (msg.dest == "fromPd_quit")
                    {
                        running = false;
                    }
                    break;

                case pd::FLOAT:
                    std::cout << "FLOAT: " << msg.num << "\n";
                    if (msg.dest == "fromPd_filterfreq")
                    {
                        filter.left.setFrequency(msg.num);
                        filter.right.setFrequency(msg.num);
                    }
                    else if (msg.dest == "fromPd_filtertype")
                    {
                        // (filterType is just an array containing the available filter types.)
                        filter.left.setState(filterType[(unsigned int)msg.num]);
                        filter.right.setState(filterType[(unsigned int)msg.num]);
                    }
                    else if (msg.dest == "fromPd_filtergain")
                    {
                        filter.left.setGain(msg.num);
                        filter.right.setGain(msg.num);
                    }
                    else if (msg.dest == "fromPd_filterbw")
                    {
                        filter.left.setBandwidth(msg.num);
                        filter.right.setBandwidth(msg.num);
                    }
                    break;

                default:
                    std::cout << "Unknown Pd message.\n";
                    std::cout << "Type: " << msg.type << ", " << msg.dest << "\n";
                    break;
            }
        }
    }
}

Once the test has ended by banging the stop_test button on the control patch, cleanup is as follows:

int main(int argc, const char * argv[])
{
    <snip> // As above.

    pd.unsubscribeAll();
    pd.computeAudio(false);
    pd.closePatch(testData.pdPatch);

    AudioOutputUnitStop(testData.outputUnit);
    AudioUnitUninitialize(testData.outputUnit);
    AudioComponentInstanceDispose(testData.outputUnit);

    delete[] testData.pdBuffer;

    return 0;
}

The raw synth tone in the receiver patch used as the test signal is actually built with the PolyBLEP oscillator I made and discussed in a previous post. So it’s also possible (and very easy) to compile custom Pure Data externals into libpd, and that’s pretty awesome! Here is a demonstration of what I’ve been talking about — testing a state-variable filter on a raw synth tone:

Pure Data & libpd Interactive Demo from Christian on Vimeo.

Dynamics Processing: Compressor/Limiter, part 2

In part 1 I detailed how I built the envelope detector that I will now use in my Unity compressor/limiter. To reiterate, the envelope detector extracts the amplitude contour of the audio that will be used by the compressor to determine when to compress the signal’s gain. The response of the compressor is determined by the attack time and the release time of the envelope, with higher values resulting in a smoother envelope, and hence, a gentler response in the compressor.

The compressor script is a MonoBehaviour component that can be attached to any GameObject. Here are the fields and corresponding inspector GUI:

public class Compressor : MonoBehaviour
{
    [AudioSlider("Threshold (dB)", -60f, 0f)]
    public float threshold = 0f;		// in dB
    [AudioSlider("Ratio (x:1)", 1f, 20f)]
    public float ratio = 1f;
    [AudioSlider("Knee", 0f, 1f)]
    public float knee = 0.2f;
    [AudioSlider("Pre-gain (dB)", -12f, 24f)]
    public float preGain = 0f;			// in dB, amplifies the audio signal prior to envelope detection.
    [AudioSlider("Post-gain (dB)", -12f, 24f)]
    public float postGain = 0f;			// in dB, amplifies the audio signal after compression.
    [AudioSlider("Attack time (ms)", 0f, 200f)]
    public float attackTime = 10f;		// in ms
    [AudioSlider("Release time (ms)", 10f, 3000f)]
    public float releaseTime = 50f;		// in ms
    [AudioSlider("Lookahead time (ms)", 0, 200f)]
    public float lookaheadTime = 0f;	// in ms

    public ProcessType processType = ProcessType.Compressor;
    public DetectionMode detectMode = DetectionMode.Peak;

    private EnvelopeDetector[] m_EnvelopeDetector;
    private Delay m_LookaheadDelay;

    private delegate float SlopeCalculation (float ratio);
    private SlopeCalculation m_SlopeFunc;
    
    // Continued...
Compressor/Limiter Unity inspector GUI.

Compressor/Limiter Unity inspector GUI.

Compressor/Limiter Unity inspector GUI.

Compressor/Limiter Unity inspector GUI.

 

 

 

 

 

 

 

 

The two most important parameters for a compressor are the threshold and the ratio values. When a signal exceeds the threshold, the compressor reduces the level of the signal by the given ratio. For example, if the threshold is -2 dB with a ratio of 4:1 and the compressor encounters a signal peak of +2 dB, the gain reduction will be 3 dB, resulting in the signal’s new level of -1dB. The ratio is just a percentage, so a 4:1 ratio means that the signal will be reduced by 75% (1 – 1/4 = 0.75). The difference between the threshold and the signal peak (which is 4 dB in this example) is scaled by the ratio to arrive at the 3 dB reduction (4 * 0.75 = 3). When the ratio is ∞:1, the compressor is turned into a limiter. The compressor’s output can be visualized by a plot of amplitude in vs. amplitude out:

Plot of amplitude in vs. amplitdue out of a compressor with 4:1 ratio.

Plot of amplitude in vs. amplitdue out of a compressor with 4:1 ratio.

When the ratio is ∞:1, the resulting amplitude after the threshold would be a straight horizontal line in the above plot, effectively preventing any levels from exceeding the threshold. It can easily be seen how this then would exhibit the behavior of a limiter. From these observations, we can derive the equations we need for the compressor.

compressor gain = slope * (threshold – envelope value) if envelope value >= threshold, otherwise 0

slope = 1 – (1 / ratio), or for limiting, slope = 1

All amplitude values are in dB for these equations. We saw both of these equations earlier in the example I gave, and both are pretty straightforward. These elements can now be combined to make up the compressor/limiter. The Awake method is called as soon as the component is initialized in the scene.

 

void Awake ()
{
    if (processType == ProcessType.Compressor) {
        m_SlopeFunc = CompressorSlope;
    } else if (processType == ProcessType.Limiter) {
        m_SlopeFunc = LimiterSlope;
    }

    // Convert from ms to s.
    attackTime /= 1000f;
    releaseTime /= 1000f;

    // Handle stereo max number of channels for now.
    m_EnvelopeDetector = new EnvelopeDetector[2];
    m_EnvelopeDetector[0] = new EnvelopeDetector(attackTime, releaseTime, detectMode, sampleRate);
    m_EnvelopeDetector[1] = new EnvelopeDetector(attackTime, releaseTime, detectMode, sampleRate);
}

Here is the full compressor/limiter code in Unity’s audio callback method. When placed on a component with the audio listener, the data array will contain the audio signal prior to being sent to the system’s output.

void OnAudioFilterRead (float[] data, int numChannels)
{
    float postGainAmp = AudioUtil.dB2Amp(postGain);

    if (preGain != 0f) {
        float preGainAmp = AudioUtil.dB2Amp(preGain);
        for (int k = 0; k < data.Length; ++k) {
            data[k] *= preGainAmp;
        }
    }

    float[][] envelopeData = new float[numChannels][];

    if (numChannels == 2) {
        float[][] channels;
        AudioUtil.DeinterleaveBuffer(data, out channels, numChannels);
        m_EnvelopeDetector[0].GetEnvelope(channels[0], out envelopeData[0]);
        m_EnvelopeDetector[1].GetEnvelope(channels[1], out envelopeData[1]);
        for (int n = 0; n < envelopeData[0].Length; ++n) {
            envelopeData[0][n] = Mathf.Max(envelopeData[0][n], envelopeData[1][n]);
        }
    } else if (numChannels == 1) {
        m_EnvelopeDetector[0].GetEnvelope(data, out envelopeData[0]);
    } else {
        // Error...
    }

    m_Slope = m_SlopeFunc(ratio);

    for (int i = 0, j = 0; i < data.Length; i+=numChannels, ++j) {
        m_Gain = m_Slope * (threshold - AudioUtil.Amp2dB(envelopeData[0][j]));
        m_Gain = Mathf.Min(0f, m_Gain);
        m_Gain = AudioUtil.dB2Amp(m_Gain);
        for (int chan = 0; chan < numChannels; ++chan) {
            data[i+chan] *= (m_Gain * postGainAmp);
        }
    }
}

And quickly, here is the helper method for deinterleaving a multichannel buffer:

public static void DeinterleaveBuffer (float[] source, out float[][] output, int sourceChannels)
{
    int channelLength = source.Length / sourceChannels;

    output = new float[sourceChannels][];

    for (int i = 0; i < sourceChannels; ++i) {
        output[i] = new float[channelLength];

        for (int j = 0; j < channelLength; ++j) {
            output[i][j] = source[j*sourceChannels+i];
        }
    }
}

First off, there are a few utility functions that I included in the component that converts between linear amplitude and dB values that we can see in the function above. Pre-gain is applied to the audio signal prior to extracting the envelope. For multichannel audio, Unity unfortunately gives us an interleaved buffer, so this needs to be deinterleaved before sending it to the envelope detector (recall that the detector uses a recursive filter and thus has state variables. This could of course be handled differently in the envelope detector, but it’s simpler to work on single continuous data buffers).

When working with multichannel audio, each channel will have a unique envelope. These could of course be processed separately, but this will result in the relative levels between the channels to be disturbed. Instead, I take the maximum envelope value and use that for the compressor. Another option would be to take the average of the two.

I then calculate the slope value based on whether the component is set to compressor or limiter mode (via a function delegate). The following loop is just realizing the equations posted earlier, and converting the dB gain value to linear amplitude before applying it to the audio signal along with post-gain.

This completes the compressor/limiter component. However, there are two important elements missing: soft knee processing, and lookahead. From the plot earlier in the post, we see that once the signal reaches the threshold, the compressor kicks in rather abruptly. This point is called the knee of the compressor, and if we want this transition to happen more gently, we can interpolate within a zone around the threshold.

It’s common, especially in limiters, to have a lookahead feature that compensates for the obvious lag of the envelope detector. In other words, when the attack and release times are non-zero, the resulting envelope lags behind the audio signal as a result of the filtering. The compressor/limiter will actually miss attenuating the peaks in the signal that it needs to because of this lag. That’s where lookahead comes in. In truth, it’s a bit of a misnomer because we can obviously not see into the future of an audio signal, but we can delay the audio to achieve the same effect. This means that we extract the envelope as normal, but delay the audio output so that the compressor gain value lines up with the audio peaks that it is meant to attenuate.

I will be implementing these two remaining features in a future post.

Dynamics processing: Compressor/Limiter, part 1

Lately I’ve been busy developing an audio-focused game in Unity, whose built-in audio engine is notorious for being extremely basic and lacking in features. (As of this writing, Unity 5 has not yet been released, in which its entire built-in audio engine is being overhauled). For this project I have created all the DSP effects myself as script components, whose behavior is driven by Unity’s coroutines. In order to have slightly more control over the final mix of these elements, it became clear that I needed a compressor/limiter. This particular post is written with Unity/C# in mind, but the theory and code is easy enough to adapt to other uses. In this first part we’ll be looking at writing the envelope detector, which is needed by the compressor to do its job.

An envelope detector (also called a follower) extracts the amplitude envelope from an audio signal based on three parameters: an attack time, release time, and detection mode. The attack/release times are fairly straightforward, simply defining how quickly the detection responds to rising and falling amplitudes. There are typically two modes of calculating the envelope of a signal: by its peak value or its root mean square value. A signal’s peak value is just the instantaneous sample value while the root mean square is measured over a series of samples, and gives a more accurate account of the signal’s power. The root mean square is calculated as:

rms = sqrt ( (1/n) * (x12 + x22 + … + xn2) ),

where n is the number of data values. In other words, we sum together the squares of all the sample values in the buffer, find the average by dividing by n, and then taking the square root. In audio processing, however, we normally bound the sample size (n) to some fixed number (called windowing). This effectively means that we calculate the RMS value over the past n samples.

(As an aside, multiplying by 1/n effectively assigns equal weights to all the terms, making it a rectangular window. Other window equations can be used instead which would favor terms in the middle of the window. This results in even greater accuracy of the RMS value since brand new samples (or old ones at the end of the window) have less influence over the signal’s power.)

Now that we’ve seen the two modes of detecting a signal’s envelope, we can move on to look at the role of the attack/release times. These values are used in calculating coefficients for a first-order recursive filter (also called a leaky integrator) that processes the values we get from the audio buffer (through one of the two detection methods). Simply stated, we get the sample values from the audio signal and pass them through a low-pass filter to smooth out the envelope.

We calculate the coefficients using the time-constant equation:

g = e ^ ( -1 / (time * sample rate) ),

where time is in seconds, and sample rate in Hz. Once we have our gain coefficients for attack/release, we put them into our leaky integrator equation:

out = in + g * (out – in),

where in is the input sample we detected from the incoming audio, g is either the attack or release gain, and out is the envelope sample value. Here it is in code:

public void GetEnvelope (float[] audioData, out float[] envelope)
{
    envelope = new float[audioData.Length];

    m_Detector.Buffer = audioData;

    for (int i = 0; i < audioData.Length; ++i) {
        float envIn = m_Detector[i];

        if (m_EnvelopeSample < envIn) {
            m_EnvelopeSample = envIn + m_AttackGain * (m_EnvelopeSample - envIn);
        } else {
            m_EnvelopeSample = envIn + m_ReleaseGain * (m_EnvelopeSample - envIn);
        }

        envelope[i] = m_EnvelopeSample;
    }
}

(Source: code is based on “Envelope detector” from http://www.musicdsp.org/archive.php?classid=2#97, with detection modes added by me.)

The envelope sample is calculated based on whether the current audio sample is rising or falling, with the envIn sample resulting from one of the two detection modes. This is implemented similarly to what is known as a functor in C++. I prefer this method to having another branching structure inside the loop because among other things, it’s more extensible and results in cleaner code (as well as being modular). It could be implemented using delegates/function pointers, but the advantage of a functor is that it retains its own state, which is useful for the RMS calculation as we will see. Here is how the interface and classes are declared for the detection modes:

public interface IEnvelopeDetection
{
    float[] Buffer { set; get; }
    float this [int index] { get; }

    void Reset ();
}

We then have two classes that implement this interface, one for each mode:

A signal’s peak value is the instantaneous sample value while the root mean square is measured over a series of samples, and gives a more accurate account of the signal’s power.

public class DetectPeak : IEnvelopeDetection
{
    private float[] m_Buffer;

    /// <summary>
    /// Sets the buffer to extract envelope data from. The original buffer data is held by reference (not copied).
    /// </summary>
    public float[] Buffer
    {
        set { m_Buffer = value; }
        get { return m_Buffer; }
    }

    /// <summary>
    /// Returns the envelope data at the specified position in the buffer.
    /// </summary>
    public float this [int index]
    {
        get { return Mathf.Abs(m_Buffer[index]); }
    }

    public DetectPeak () {}
    public void Reset () {}
}

This particular class involves a rather trivial operation of just returning the absolute value of a signal’s sample. The RMS detection class is more involved.

/// <summary>
/// Calculates and returns the root mean square value of the buffer. A circular buffer is used to simplify the calculation, which avoids
/// the need to sum up all the terms in the window each time.
/// </summary>
public float this [int index]
{
    get {
        float sampleSquared = m_Buffer[index] * m_Buffer[index];
        float total = 0f;
        float rmsValue;

        if (m_Iter < m_RmsWindow.Length-1) {
            total = m_LastTotal + sampleSquared;
            rmsValue = Mathf.Sqrt((1f / (index+1)) * total);
        } else {
            total = m_LastTotal + sampleSquared - m_RmsWindow.Read();
            rmsValue = Mathf.Sqrt((1f / m_RmsWindow.Length) * total);
        }

        m_RmsWindow.Write(sampleSquared);
        m_LastTotal = total;
        m_Iter++;

        return rmsValue;
    }
}

public DetectRms ()
{
    m_Iter = 0;
    m_LastTotal = 0f;
    // Set a window length to an arbitrary 128 for now.
    m_RmsWindow = new RingBuffer<float>(128);
}

public void Reset ()
{
    m_Iter = 0;
    m_LastTotal = 0f;
    m_RmsWindow.Clear(0f);
}

The RMS calculation in this class is an optimization of the general equation I stated earlier. Instead of continually summing together all the  values in the window for each new sample, a ring buffer is used to save each new term. Since there is only ever 1 new term to include in the calculation, it can be simplified by storing all the squared sample values in the ring buffer and using it to subtract from our previous total. We are just left with a multiply and square root, instead of having to redundantly add together 128 terms (or however big n is). An iterator variable ensures that the state of the detector remains consistent across successive audio blocks.

In the envelope detector class, the detection mode is selected by assigning the corresponding class to the ivar:

public class EnvelopeDetector
{
    protected float m_AttackTime;
    protected float m_ReleaseTime;
    protected float m_AttackGain;
    protected float m_ReleaseGain;
    protected float m_SampleRate;
    protected float m_EnvelopeSample;

    protected DetectionMode m_DetectMode;
    protected IEnvelopeDetection m_Detector;

    // Continued...
public DetectionMode DetectMode
{
    get { return m_DetectMode; }
    set {
        switch(m_DetectMode) {
            case DetectionMode.Peak:
                m_Detector = new DetectPeak();
                break;

            case DetectionMode.Rms:
                m_Detector = new DetectRms();
                break;
        }
    }
}

Now that we’ve looked at extracting the envelope from an audio signal, we will look at using it to create a compressor/limiter component to be used in Unity. That will be upcoming in part 2.

AdVerb: Building a Reverb Plug-In Using Modulating Comb Filters

Some time ago, I began exploring the early reverb algorithms of Schroeder and Moorer, whose work dates back all the way to the 1960s and 70s respectively.  Still their designs and theories inform the making of algorithmic reverbs today.  Recently I took it upon myself to continue experimenting with the Moorer design I left off with in an earlier post.  This resulted in the complete reverb plug-in “AdVerb”, which is available for free in downloads.  Let me share what went into designing and implementing this effect.

One of the foremost challenges in basing a reverb design on Schroeder or Moorer is that it tends to sound a little metallic because with the number of comb filters suggested, the echo density doesn’t build up fast or dense enough.  The all-pass filters in series that come after the comb filter section helps to diffuse the reverb tail, but I found that the delaying all-pass filters added a little metallic sound of their own.  One obvious way of overcoming this is to add more comb filters (today’s computers can certainly handle it).  More importantly, however, the delay times of the comb filters need to be mutually prime so that their frequency responses don’t overlap, which would result in increased beating in the reverb tail.

To arrive at my values for the 8 comb filters I’m using, I wrote a simple little script that calculated the greatest common divisor between all the delay times I chose and made sure that the results were 1.  This required a little bit of tweaking in the numbers, as you can imagine finding 8 coprimes is not as easy as it sounds, especially when trying to keep the range minimal between them.  It’s not as important for the two all-pass filters to be mutually prime because they are in series, not in parallel like the comb filters.

I also discovered, after a number of tests, that the tap delay used to generate the early reflections (based on Moorer’s design) was causing some problems in my sound.  I’m still a bit unsure as to why, though it could be poorly chosen tap delay times or something to do with mixing, but it was enough so that I decided to discard the tap delay network and just focus on comb filters and all-pass filters.  It was then that I took an idea from Dattorro and Frenette who both showed how the use of modulated comb/all-pass filters can help smear the echo density and add warmth to the reverb.  Dattorro is responsible for the well-known plate reverbs that use modulating all-pass filters in series.

The idea behind a modulated delay line is that some oscillator (usually a low-frequency sine wave) modulates the delay value according to a frequency rate and amplitude.  This is actually the basis for chorusing and flanging effects.  In a reverb, however, the values need to be kept very small so that the chorusing effect will not be evident.

I had fun experimenting with these modulated delay lines, and so I eventually decided to modulate one of the all-pass filters as well and give control of it to the user, which offers a great deal more fun and crazy ways to use this plug-in.  Let’s take a look at the modulated all-pass filter (the modulated comb filter is very similar).  We already know what an all-pass filter looks like, so here’s just the modulated delay line:

Modulated all-pass filter.

Modulated all-pass filter.

The oscillator modulates the value currently in the delay line that we then use to interpolate, resulting in the actual value.  In code it looks like this:

double offset, read_offset, fraction, next;
size_t read_pos;

offset = (delay_length / 2.) * (1. + sin(phase) * depth);
phase += phase_incr;
if (phase > TWO_PI) phase -= TWO_PI;
if (offset > delay_length) offset = delay_length;

read_offset = ((size_t)delay_buffer->p - (size_t)delay_buffer->p_head) / sizeof(double) - offset;
if (read_offset < 0) {
    read_offset = read_offset + delay_length;
} else if (read_offset > delay_length) {
    read_offset = read_offset - delay_length;
}

read_pos = (size_t)read_offset;
fraction = read_offset - read_pos;
if (read_pos != delay_length - 1) {
    next = *(delay_buffer->p_head + read_pos + 1);
} else {
    next = *delay_buffer->p_head;
}

return *(delay_buffer->p_head + read_pos) + fraction * (next - *(delay_buffer->p_head + read_pos));

In case that looks a little daunting, we’ll step through the C code (apologies for the pointer arithmetic!).  At the top we calculate the offset using the delay length in samples as our base point.  The following lines are easily seen as incrementing and wrapping the phase of the oscillator as well as capping the offset to the delay length.

The next line calculates the current position in the buffer from the current position pointer, p, and the buffer head, p_head.  This is accomplished by casting the pointer addresses to integral values and dividing by the size of the data type of each buffer element.  The read_offset position will determine where in the delay buffer we read from, so it needs to be clamped to the buffer’s length as well.

The rest is simply linear interpolation (albeit with some pointer arithmetic: delay_buffer->p_head + read_pos + 1 is equivalent to delay_buffer[read_pos + 1]).  Once we have our modulated delay value, we can finish processing the all-pass filter:

delay_val = get_modulated_delay_value(allpass_filter);

// don't write the modulated delay_val into the buffer, only use it for the output sample
*delay_buffer->p = sample_in + (*delay_buffer->p * allpass_filter->g);
sample_out = delay_val - (allpass_filter->g * sample_in);

The final topology of the reverb is given below:

Topology of the AdVerb plug-in.

Topology of the AdVerb plug-in.

The pre-delay is implemented by a simple delay line, and the low-pass filters are of the one-pole IIR variety.  Putting the LPFs inside the comb filters’ feedback loops simulates the absorption of energy that sound undergoes as it comes in contact with surfaces and travels through air.  This factor can be controlled with a damping parameter in the plug-in.

The one-pole moving-average filter is there for an extra bit of high frequency roll-off, and I chose it because this particular filter is an FIR type and has linear phase so it won’t add further disturbance to the modulated samples entering it.  The last (normal) all-pass filter in the series serves to add extra diffusion to the reverb tail.

Here are some short sound samples using a selection of presets included in the plug-in:

Piano, “Medium Room” preset

The preceding sample demonstrates a normal reverb setting.  Following are a few samples that demonstrate a couple of subtle and not-so-subtle effects:

Piano, “Make it Vintage” preset

Piano, “Bad Grammar” preset

Flute, “Shimmering Tail” preset

Feel free to get in touch regarding any questions or comments on “AdVerb“.

The Different “Sides” of Convolution

We all know how much convolution is used in DSP, and what a significant part it has in making many effects and analysis techniques a reality.  Often we hear about FFT, or fast convolution, but in actuality these algorithms are only faster than straight convolution with a kernel length (of the impulse response) greater than around 60 – 64 or so.  Anything shorter than that is best handled with normal convolution, which can be implemented using either an input side or output side algorithm.

The input side algorithm is normally the one that we learn first (at least it was for me, and judging from many texts and books on it, it seems to be common).  However, the output side algorithm is perhaps slightly easier to code, and we’ll see why later on.  For this post I’m going to evaluate the effects each algorithm has on the resulting audio signal because each one could have an advantage in some situations over the other.  I’ll also share a little trick that will speed up the convolution process for certain types of filters.

The output side algorithm performs the convolution, as one might expect, from the viewpoint of the output (or the result) of the operation, while the input side algorithm implements convolution from the viewpoint of the input signal.  To look at it another way, recall that convolving an input signal with an impulse response results in an output signal that is the length of the input + the length of the impulse response – 1.  With the input side algorithm, this “extra” bit of length is appended to the resulting signal, and is probably the easiest one to theoretically grasp.  The output side algorithm looks at what points from the input we need in order to get our resulting signal.

The reason why the output side method feels more natural to code is that we often write in terms of the output or result: y(n) = ____ — the output is equal to some kind of expression, for example.  Approaching convolution this way requires input samples on the negative side (x[-2], x[-1], etc.), which of course don’t exist.  To get around that little problem, one common way is to implement a delay line that contains all zeroes at first, then shifting in the input signal as we go, doing the calculating with the impulse response and the delay line instead of direclty with the input.

I’ve been busy redesigning and overhauling my DSP filter class, which has entailed some experimentation with these two convolution methods on FIR filters.  Here we can see visually the difference between them.  First up is the output side convolution of a windowed-sinc FIR filter using a Kaiser window of order 162 on a simple triangle wave (using test oscillators is a good way to visually see what’s happening).

Output side convolution on a triangle wave.

Output side convolution on a triangle wave.

I purposely made the filter kernel (impulse response) fairly long so it would illustrate the delay and shift that’s happening at the start of the signal due to the output side algorithm.  Compare that with the image of the input side algorithm using exactly the same filter and order.

Input side convolution on a triangle wave.

Input side convolution on a triangle wave.

Huge difference.  It’s clearly audible as well in this test signal (samples below).  With the output side algorithm there is a clear popping sound as the audio abruptly starts in contrast to the much smoother beginning of the test signal filtered with the input side algorithm.

Here you can hear the difference (to hear it clearly you may have to download the files and listen in an audio editor to avoid the glitchy start of web browser plug-ins):

Triangle wave filtered with output side

Triangle wave filtered with input side

Now to return to the point about the output side method being perhaps a little more natural to code.  The little wrinkle we face in implementing the input side algorithm is that with block processing (which is so common in DSP), how do we account for the longer output as a result of the convolution?  In many cases we don’t have control over the size of the audio buffers we’re given or have to work with.  The solution is to use the overlap-add method.

It’s really not more difficult than the output side algorithm.  All we have to do is calculate the convolution fully (into our own internal buffer if we have to), save the extra length at the end (which will be equal to the impulse response length), then add that to the start of the next block.  Rinse and repeat.  Here is C++ code that implements both of these convolution methods.

Output side convolution process.

Output side convolution process.

Input side convolution process.

Input side convolution process.

Don’t worry, I didn’t forget to share the trick I mentioned at the beginning.  Many FIR filters have linear phase response, which means that their kernels are symmetrical.  That provides us a great opportunity to eliminate extra calculations that aren’t needed.  So notice in the above code each ‘j’ loop (the kernel or impulse response loop) only traverses half the kernel length, as the value of input[i] * mKernel[j] is the same as input[i] * mKernel[mKernelLength-j-1].  At the end of the kernel’s loop we calculate the midpoint value.

Again, the calculations involving symmetry is perhaps easier to see in the output side algorithm, because we naturally gravitate towards expressions that result in a single output value.  If you work a small convolution problem out on paper, however, it will help you see the input side algorithm and why it works.

Audio Resampling: Part 2

This post has been edited to clarify some of the details of implementing the polyphase resampler [May 14, 2013].

Time to finish up this look at resampling.  In Part 1 we introduced the need for resampling to avoid aliasing in signals, and its implementation by windowed sinc FIR filters.  This is a costly operation, however, especially for real-time processing.  Let’s consider the case of upsampling-downsampling a 1 second audio signal at a sampling rate of 44.1 kHz and a resampling factor of 4X.  Going about this with the brute-force method that we saw in Part 1 would result in first upsampling the signal by 4X.  This results in a buffer size that has now grown to be 4 x 44100 = 176,400 that we now have to filter, which will obviously take roughly 4 times as long to compute.  And not only once, but twice, because the decimation filter also operates at this sample rate.  The way around this is to use polyphase interpolating filters.

A polyphase filter is a set of subfilters where the filter kernel has been split up into a matrix with each row representing a subfilter.  The input samples are passed into each subfilter that are then summed together to produce the output.  For example, given the impulse response of the filter

transf1

we can separate it into two subfilters, E0 and E1

transf2

transf3

where E0 contains the even-numbered kernel coefficients and E1 contains the odd ones.  We can then express H(z) as

fullequation

This can of course be extended for any number of subfilters.  In fact, the number of subfilters in the polyphase interpolating/decimating filters is exactly the same as the resampling factor.  So if we’re upsampling-downsampling by a factor of 4X, we use 4 subfilters.  However, we still have the problem of filtering on 4 times the number of samples as a result of the upsampling.  Here is where the Noble Identity comes in.  It states that for multirate signal processing, filtering before upsampling and filtering after downsampling is equivalent to filtering after upsampling and before downsampling.

Noble Identities for upsampling and downsampling.

Noble Identities for upsampling and downsampling.

When you think about it, it makes sense.  Recall that upsampling involves zero-insertion between the existing samples.  These 0 values, when passed through the filter, simply evaluate out to 0 and have no effect on the resulting signal, so they are wasted calculations.  We are now able to save a lot of computational expense by filtering prior to upsampling the signal, and then filtering after downsampling.  In other words, the actual filtering takes place at the original sample rate.  The way this works in with the polyphase filter is quite clever: through a commutator switch.

Let’s take the case of decimation first because it’s the easier one to understand.  Before the signal’s samples enter the polyphase filter, a commutator selects every Mth sample (M is the decimation factor) to pass into the filter while discarding the rest.  The results of each subfilter is summed together to produce the signal, back to its original sample rate and without aliasing components.  This is illustrated in the following diagram:

Polyphase decimator with a commutator switch that selects the input.

Polyphase decimator with a commutator switch that selects the input.

Interpolation works much the same, but obviously on the other end of the filter.  Each input sample from the signal passes through the polyphase filter, but instead of summing together the subfilers, a commutator switch selects the outputs of the subfilters that make up the resulting upsampled signal.  Think of it this way: one sample passes into each of the subfilters that then results in L outputs (L being the interpolation factor and the number of subfilters).  The following diagram shows this:

Polyphase interpolating filter with a commutator switch.

Polyphase interpolating filter with a commutator switch that selects the output.

We now have a much more efficient resampling filter.  There are other methods that exist as well to implement a resampling filter, including Fast Fourier Transform, which is a fast and efficient way of doing convolution, and would be a preferred method of implementing FIR filters.  At lower orders however, straight convolution is just as fast (if not even slightly faster at orders less than 60 or so) than FFT; the huge gain in efficiency really only occurs with a kernel length greater than 80 – 100 or so.

Before concluding, let’s look at some C++ code fragments that implement these two polyphase structures.  Previously I had done all the work inside a single method that contained all the for loops to implement the convolution.  Since we’re dealing with a polyphase structure, it naturally follows that the code should be refactored into smaller chunks since each filter branch can be throught of as an individual filter.

First, given the prototype filter’s kernel, we break it up into the subfilter branches.  The number of subfilters (branches) we need is simply equal to the resampling factor.  Each filter branch will then have a length equal to the prototype filter’s kernel length divided by the factor, then +1 to take care of rounding error.  i.e.

branch order = (prototype filter kernel length / factor) + 1

The total order of the polyphase structure will then be equal to the branch order x number of branches, which will be larger than the prototype kernel, but any extra elements should be initialized to 0 so they won’t affect the outcome.

The delay line, z, for the interpolator will have a length equal to the branch order.  Again, each branch can be thought of as a separate filter.  First, here is the decimating resampling code:

 

decim_code_2

Example of decimating resampler code.

As can be seen, calculating each polyphase branch is handled by a separate object with its own method of calculating the subfilter (processDownsample).  We index the input signal with variable M, advances at the rate of the resampling factor.  The gain adjust can be more or less ignored depending on how the resampling is implemented.  In my case, I have precalculated the prototype filter kernels to greatly improve efficiency.  However, the interpolation process decreases the level of the signal by an amount equal to the resampling factor in decibels.  In other words, if our factor is 3X, we need to amplify the interpolated signal by 3dB.  I’ve done this by amplifying the prototype filter kernel so I don’t need to adjust the gain during interpolation.  But this means I need to compensate for that in decimation by reducing the level of the signal by the same amount.

Here is the interpolator code:

Example interpolation resampling code.

Example of interpolation resampler code.

As we can see, it’s quite similar to the decimation code, except that the output selector requires an additional for loop to distribute the results of the polyphase branches.  Similarly though, it uses the same polyphase filter object to calculate each filter branch, using the delay line as input instead of the input signal directly.  Here is the code for the polyphase branches:

Code implementing the polyphase branches.

Code implementing the polyphase branches.

Again, quite similar, but with a few important differences.  The decimation/downsampling MACs the input sample by each kernel value whereas interpolation/upsampling MACs the delay line with the branch kernel.

Hopefully this clears up a bit of confusion regarding the implementation of the polyphase filter.  Though this method splits up and divides the tasks of calculating the resampling into various smaller objects than before, it is much easier to understand and maintain.

Resampling, as we have seen, is not a cheap operation, especially if a strong filter is required.  However, noticeable aliasing will render any audio unusable, and once it’s in the signal it cannot be removed.  Probably the best way to avoid aliasing is to prevent it in the first place by using band-limited oscillators or other methods to keep all frequencies below the Nyquist limit, but this isn’t always possible as I pointed out in Part 1 with ring modulation, distortion effects, etc.  There is really no shortage of challenges to deal with in digital audio!

Audio Resampling: Part 1

Resampling in digital audio has two main uses.  One is to convert audio into a sampling rate needed by a particular system or engine (e.g. converting 48kHz audio to the required 44.1kHz required by CDs).  The second is to avoid aliasing during signal processing by raising the Nyquist limit.  I will be discussing the latter.

Lately I’ve been very busy working on improving and enhancing the sound of ring modulation for a fairly basic plug-in being developed by AlgoRhythm Audio (coming soon).  I say basic becuase as far as ring modulation goes, there are few DSP effects that are simpler in theory and in execution.  Simply take some input signal, multiply it by a carrier signal (usually some kind of oscillator like a sine wave), and we have ring modulation.  The problem that arises, however, and how this connects in with resampling, is that this creates new frequencies in the resulting output that were not present in either signal prior to processing.  These new frequencies created could very well violate the Nyquist limit of the current sampling rate during processing, and that leads us to resampling as a way to clamp down on aliasing frequencies that can be introduced as a result.

Aliasing is an interesting phenomenon that occurs in digital audio, and in every sense of the word is an undesired noise that we need to make sure does not pollute our audio.  There are many resources around that go into more detail on aliasing, but I will give a brief overview of it with some audio and visual samples.

Aliasing occurs when there are frequencies present in a signal that are greater than the Nyquist limit (half of the sampling rate).  What happens in such a case is that the sampling rate is not high enough to properly capture (sample) the high frequency of the signal, and so the frequency “folds over” and creates aliases that are mirror images of the original frequency.  Here, for example, is a square wave at 4000Hz created using 20 harmonics at a sampling rate of 44.1kHz (keep in mind that 4000Hz is the fundamental frequency, and that square waves contain many additional frequencies above that depending on how many harmonics were used to create it, so in this case Nyquist is still being violated):

4000Hz square wave with aliasing

Notice the low tone below the actual 4000Hz frequency.  Here is the resulting waveform of this square wave that shows us visually that we’re not sampling fast enough to accurately reproduce the waveform.  Notice the inconsistencies in the waveform.

squarewave_alias

Now, going to the extreme a bit, here is the same square wave sampled at a rate of 192kHz.

4000Hz square wave with no aliasing

It’s a pure 4000Hz square wave tone.  Examining the waveform of this square wave shows us that the sampling rate was more that adequate to reproduce this signal digitally:

square_noalias

Not all signal processing effects are susceptible to aliasing, and certainly not to the same degree.  Because ring modulation produces additional inharmonic frequencies, it is a prime example of a process that is easily affected by aliasing.  Others include distortion and various other kinds of modulation techniques (especially when taken to the extreme).  However, ring modulation with a sine wave is generally safe as long as the frequency of the sine wave is kept  low enough because sine waves have no harmonics to them, only the fundamental.  Introducing other wavetypes into this process, however, can quite easily bring about aliasing.

Here is an example of ring modulation with a sawtooth wave sweeping up from about 200Hz to 5000Hz.  As it glides up, you will be able to hear the aliasing kicking in at around 0:16 or so.  The example with no aliasing has been upsampled by 3X before processing, then downsampled by 3X back to its original rate.

Ring modulation with a sawtooth sweep, with aliasing

Ring modulation with sawtooth sweep, with no aliasing

So how does upsampling and downsampling work?  In theory, and even to some extent in practice, it’s very straightforward.  The issue, as we will see, is in making it efficient and fast.  In DSP we’re always concerning ourselves with speedy execution times to avoid latency or audio dropouts or running out of memory, etc.

To upsample, we insert 0-valued samples in between every L-th (our upsampling factor) original sample.  (i.e. upsampling by 3X, [1, 2, 3, 4] becomes [1,0,0,2,0,0,3,0,0…])  However, this introduces aliasing into our audio so we need to interpolate these values so that they “join” the sample values of the original waveform.  This is accomplished by using an interpolating low-pass filter.  This entire process is known as interpolation.

Dowsampling is much the same.  We remove every M-th (our downsampling factor) sample from the original signal.  (i.e. downsampling by 2X, [1,2,3,4,5] becomes [1,3,5…])  This process does not introduce aliasing, but we do need to make sure the Nyquist limit is adhered to at the new sampling rate by low-pass filtering with a cutoff frequency at the new Nyquist rate.  This entire process is known as decimation.

Fairly straightforward.  Proceeding with this method, however, would be known as brute force — generally not a good way to go.  The reason why becomes clearer when we consider what kind of low-pass filter we need for this operation.  The ideal filter would be one that would brick-wall attenuate all frequencies higher than Nyquist and leave everything else untouched (thus preserving all the frequencies and tonal content of our original audio).  This is, alas, impossible, as it would require an infinitely long filter kernel.  The function that would implement this ideal filter is the rectangular function.

rect

The rectangular function.

By taking the Fourier transform of the rectangular function we end up with the sinc function, which is given by:

y(x) = sin (x) / x, which becomes y(x) = sin (πx) / πx for signal processing.

Graph of the sinc function, sin (x) / x

Graph of the sinc function, sin (πx) / πx

The sinc functions trails on for infinity in both directions, which can be seen in the graph above, so we need to enforce bounds around it by applying a window function.  Windowing is a method of designing FIR filters by essentially “surrounding” a function (in our case the sinc) by the window, which enforces bounds so that we can properly derive a filter kernel for use in calculations.  The rectangular function shown above is a type of window, but as I mentioned, infinite slope is a deal-breaker in audio.

The Blackman window, given by the function

w(i) = 0.42 – 0.5 cos(2πi M) + 0.08 cos(4πi M),

where M is the length of the filter kernel, is a good choice for resampling because it offers a good stop-band attenuation of -74dB with good rolloff.  Putting this together with the sinc function, we can derive the filter kernel with the following formula:

Windowed-sinc kernel formula*

Windowed-sinc kernel formula*

where fc is the normalized cutoff frequency.  When i = M/2, to avoid a divide by zero, h(i) = 2fc.  K is a constant value used to achieve unity gain at zero frequency and can be ignored while calculating the kernel coefficients.  After all coefficients have been calculated, K can be found by summing together all the coefficients, h(i), and then dividing each by the resulting sum.

* Source: Smith, Steven W., “The Scientist and Engineer’s Guide to Digital Signal Processing”, Chapter 16.

Now that we have the filter design, let’s consider the properties of the FIR filter and compare them to IIR filter designs.  IIR filters give us better performance and attenuation at lower orders, meaning that they execute faster and perform better with fewer calculations than FIR filters.  However, IIR filters are still not powerful enough, even at slightly higher orders, to give us the performance we need for resampling, and trying to push IIR filters into very high orders can make them unstable and/or susceptible to quantization error due to the nature of recursion.  IIR filters also do not offer linear phase response.  FIR filters are the better choice for these reasons, but the unfortunate drawback is that they execute slowly due to being implemented by convolution.  In addition, they need to be pushed to high orders to give us the performance needed for attenuating aliasing frequencies.

However, the order of the interpolating low-pass filter can be negotiated based on the frequency content of the audio signal(s) involved.  If the audio is sufficiently oversampled, it will not contain enough frequencies near Nyquist, and as such a lower order filter can be used with a gentler rolloff without adversely affecting the audio and attenuating actual frequencies in the signal.  There are plenty of cases where we just don’t know the frequency content of the audio signals involved in processing, however, so a strong filter may be needed in these cases.  Here is a graph of a 264-order window-sinc filter (in other words, a filter kernel of length 265 including the sample x(0)):

264-order window-sinc low-pass filter frequency response

264-order window-sinc low-pass filter frequency response (cutoff frequency at 10kHz, resulting in a transition band of 882Hz)

With this in consideration, it can be easy to see that convolving a signal with a 264-order FIR filter is computationally costly for real-time processing.  There are a number of ways to improve upon this when it comes to resampling.  One is using the FFT to apply the filter.  Another interesting solution is to combine the upsampling/downsampling process into the filter itself, which can further be optimized by turning it into a polyphase filter.

The theory and implementation of a polyphase filter is a fairly long and involved topic on its own so that will be forthcoming in part 2,where we look at how to implement resampling efficiently.