With the increasing speed of todays desktop computer systems, a growing number of computationally intense tasks such as computing the Fourier transform of a sampled audio signal have become available to a broad base of users. Being a process traditionally implemented on dedicated DSP systems or rather powerful computers only available to a limited number of people, the Fourier transform can today be computed in real time on almost all average computer systems. Introducing the concept of frequency into our signal representation, this process appears to be well suited for the rather specialized application of changing the pitch of an audio signal while keeping its length constant, or changing its length while retaining its original pitch. This application is of considerable practical use in todays audio processing systems. One process that implements this has been briefly mentioned in our Time Stretching and Pitch Shifting introductory course, namely the “Phase Vocoder”. Based on the representation of a signal in the “frequency domain”, we will explicitely discuss the process of pitch shifting in this article, under the premise that time compression/expansion is analogous. Usually, pitch shifting with the Phase Vocoder is implemented by changing the time base of the signal and using a sample rate conversion on the output to achieve a change in pitch while retaining duration. Also, some implementations use explicite additive oscillator bank resynthesis for pitch shifting, which is usually rather inefficient. We will not reproduce the Phase Vocoder in its known form here, but we will use a similar process to directly change the pitch of a Fourier transformed signal in the frequency domain while retaining the original duration. The process we will describe below uses an FFT / iFFT transform pair to implement pitch shifting and automatically incorporates appropriate anti-aliasing in the frequency domain. A C language implementation of this process is provided in a black-box type routine that is easily included in an existing development setup to demonstrate the effects discussed.
1. The Short Time Fourier transform
As we have seen in our introductory course on the Fourier transform, any sampled signal can be represented by a mixture of sinusoid waves, which we called partials. Besides the most obvious manipulations that are possible based on this representation, such as filtering out unwanted frequencies, we will see that the “sum of sinusoids” model can be used to perform other interesting effects as well. It appears obvious that once we have a representation of a signal that describes it as a sum of pure frequencies, pitch shifting must be easy to implement. As we will see very soon, this is almost true.
To understand how to go about implementing pitch shifting in the “frequency domain”*, we need to take into account the obvious fact that most signals we encounter in practice, such as speech or music, are changing over time. Actually, signals that do not change over time sound very boring and do not provide a means for transmitting meaningful auditory information. However, when we take a closer look at these signals, we will see that while they appear to be changing over time in many different ways with regard to their spectrum, they remain almost constant when we only look at small “excerpts”, or “frames” of the signal that are only several milliseconds long. Thus, we can call these signals “short time stationary”, since they are almost stationary within the time frame of several milliseconds.
Because of this, it is not sensible to take the Fourier transform of our whole signal, since it will not be very meaningful: all the changes in the signals’ spectrum will be averaged together and thus individual features will not be readily observable. If we, on the other hand, split our signal into smaller “frames”, our analysis will see a rather constant signal in each frame. This way of seeing our input signal sliced into short pieces for each of which we take the DFT is called the “Short Time Fourier Transform” (STFT) of the signal.
2. Frequency Resolution Issues
To implement pitch shifting using the STFT, we need to expand our view of the traditional Fourier transform with its sinusoid basis functions a bit. In the last paragraph of our article on understanding the Fourier transform we have seen that we evaluate the Fourier transform of a signal by probing for sinusoids of known frequency and measuring the relation between the measured signal and our reference. In the article on the Fourier transform, we have chosen our reference frequencies to have an integer multiple of periods in one DFT frame. You remember that our analogy was that we have required our reference waves to use the two “nails” that are spanned by the first and last sample in our analysis “window”, like a string on a guitar that can only swing at frequencies that have their zero crossings where the string is attached to the body of the instrument. This means that the frequencies of all sinusoids we measure will be a multiple of the inverse of the analysis window length – so if our “nails” are N samples away, our STFT bins will have a spacing of sampleRate/N Hertz. As a result, this concept imposes an artificial frequency grid on our analysis by requiring the reference frequencies to be an integer multiple of our signal window in period to make them seamlessly fit into our analysis frame.
This constraint will have no consequence for the frequencies in our signal under examination that are exactly centered on our reference frequencies (since they will be a perfect fit), but since we are dealing with realworld signals we can’t expect our signal to always fulfill this requirement. In fact, the probability that one of the frequencies in our measured signal hits exactly one of our STFT bins is rather small, even more so since although it is considered short time stationary it will still slightly change over time.
So what happens to the frequencies that are between our frequency gridpoints? Well, we have briefly mentioned the effect of “smearing”, which means that they will make the largest contribution in magnitude to the bin that is closest in frequency, but they will have some of the energy “smeared” across the neighbouring bins as well. The graph below depicts how our magnitude spectrum will look like in this case.
As we can see from the above graph, when the measured signal coincides with a bin frequency it will only contribute to that bin. If it is not exactly centered on one of the bin frequencies, its magnitude will get smeared over the neighbouring bins which is why the graph in 2.2. has such a broad basis while the graph in 2.1 shows just a peak at bin 50.
The reason why I’m outlining this is that this effect is one of the key obstacles for most people who try to implement pitch shifting with the STFT. The main problem with this effect isn’t even really the magnitude spectrum, since the magnitude spectrum only tells us that a particular frequency is present in our signal. The main problem, as we will see in a minute, is the bin phase.
3. From Phase to Frequency
We have learned in our article on the Fourier transform that – with proper post-processing – it describes our signal in terms of sinusoids that have a well defined bin frequency, phase and magnitude. These three numbers alone characterize a sinusoid at any given time instant in our transform frame. We have seen that the frequency is given by the grid on which we probe the signal against our reference signal. Thus, any two bins will always be sampleRate/N Hertz away from each other in frequency. We have seen above that in the case where our measured signal coincides with the bin frequency everything is smooth – obviously it will have a frequency that is a multiple of sampleRate/N. However, what should we expect to see when it is not a multiple of sampleRate/N in frequency? Take a look at the following graph:
These two graphs look pretty normal, except that we see that the two signals obviously do not have the same frequency – the one depicted in 3.2 is of higher frequency than our sine wave in 3.1. We have announced that we will use short frames for analyzing our signals, so after splitting it up into our analysis frames it will look like this:
Our sampled signal from 3.1 that is exactly centered on a bin frequency nicely splits up into seven successive analysis frames. In each frame, the waveform starts out at zero and ends with zero. To put it more exactly: in each frame, the measured signal starts at the same point in its cycle, that is, it starts with the same phase.
Our sampled signal from 3.2 that is somewhere between two bins in frequency does not nicely split into our seven successive analysis frames. In each frame, the waveform has a clearly visible phase offset, ie. it begins at a different point in its cycle. The more off-center it is in frequency from the bin frequency, the larger this phase offset will be. So, what we see here is that while signals whose frequency is exactly on a bin frequency have the same phase offset in each frame, the phase of signals that have a frequency between two bin frequencies will have an offset that is different with each frame. So, we can deduce that a difference in phase offset between two frames denotes a deviation in frequency from our bin frequencies.
In other words: if we measure our k-th sinusoid with its bin magnitude, frequency and phase, its magnitude will denote to what extent that particular frequency is present in our signal, the frequency will take on its bin frequency and the phase will change according to its deviation from that bin frequency. Clearly, since we now know that a change in phase with each frame means a deviation in frequency from the bin frequency, we could as well use the phase offset to calculate the sinusoids’ true frequency. So, we can reduce the three numbers we get back from our post-processed analysis for each sinusoid, namely bin magnitude, bin frequency and bin phase to just magnitude and true frequency**.
Mathematically, computing the change of a parameter is known as differentiation (which means “taking the difference”, or, in the case of a function, computing the functions’ derivative), since we need to compute the difference between the current parameter value and the last parameter value, ie. how much it has changed since our last measurement. In our specific case, this parameter is the bin phase. Thus, we can say that the k-th partials’ deviation in frequency from its bin frequency is directly proportional to the derivative of the bin phase. We will use this knowledge later to compute the true frequency for that partial.
4. About The Choice of Stride
As if this wasn’t enough to consider, we also have to worry about another problem. Simply splitting the signal into successive, non-overlapping frames like we have used in our above illustration does not suffice. For several reasons, most importantly the windowing*** that needs to be done to reduce the “smearing” of the bin magnitudes, and in order to be able to uniquely discriminate the bin phase derivative without ambiguity, we need to use overlapping frames. The typical overlap factor is at least 4, ie. two adjacent windows overlap by at least 75%. Fortunately, the significance of the frame overlap on bin phase and its derivative is easy to calculate and can be subtracted out before we compute the true bin frequency. We will see how this is done in the source code example below. Actually, it is pretty unspectacular since we simply calculate how far our bin phase derivative is expected to advance for a given overlap and then subtract this offset from our phase difference prior to computing the k-th partials’ true frequency.
The other, more important thing we need to consider which is associated with this, however, is that the choice of the overlap affects the way our true partial frequencies are discriminated. If we have frames that are overlapping to a great extent, the range in which the true frequency of each of the sinusoids can vary will be larger than if we choose a smaller overlap.
To see why this is the case, let us first consider how we actually measure the bin phase. As we can see from our source code example, we use the arc tangent of the quotient of sine and cosine part (in the source code example, they are referred to as imaginary (im) and real (re) part of each bin for mathematical terminology reasons). The sign of the sine and cosine parts denote the quadrant in which the phase angle is measured. Using this knowledge we can assume that the bin phase will always be between -? and +? at any time, since this is the range of our atan2() functions’ return value. Therefore, the increase in phase between any two adjacent frames has an upper limit, with a negative angle difference denoting a negative deviation from the bin frequency and a positive angle difference denoting a positive deviation in frequency from the bin frequency. To make sure the phase difference value is centered around 0 (ie. measured as an absolute value against the origin, that is, always in the range ±? which we need in order to measure the frequency deviation) we wrap the phase difference back into our ±? interval. This is necessary since the phase offset we subtract out to make up for the overlap may cause our phase difference to be outside that interval.
Now, in the simple case that any two adjacent frames will not overlap and we know we have an interval of ±? to denote the deviation in frequency of one partial from our bin frequency, we would only be able to discriminate a frequency deviation that is ±0.5 bins since this is the maximum advance in phase between two frames we can unambiguously discriminate in this case. When the frequency of our measured signal crosses the boundary between the two adjacent STFT bins, the phase difference will wrap back to the beginning of our interval and that partials’ frequency will be far away from the actual frequency of our input sinusoid.
|Pass #1: 2411.718750 Hz (bin 112.0):
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|Pass #2: 2416.025391 Hz (bin 112.2):
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|Pass #3: 2422.270020 Hz (bin 112.49):
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We can see that when we start out at exactly the frequency of bin 112 as shown in the table of pass #1, the true frequencies for all channels that are significant (ie. have nonzero magnitude) are correctly centered on their bin frequencies as is to be expected considering we use a frequency that is a perfect fit. The 0.5 value for the magnitude of bin 111 and 113 is due to the windowing we use. When we increase the frequency of our measured signal in pass #2 to be 20% away from the 112th bin’s frequency, we see that bin 112 correctly tracks our signal, while the bin above, towards which the frequency moves, goes even farther away from the correct frequency although its magnitude increases. This is because it wraps back in its interval and actually goes into the opposite direction! This gets even more evident in pass #3 which shows that the 113th bin (which should actually be closer to the true frequency of our measured signal now according to its magnitude) goes even more into the wrong direction.
|Pass #4: 2422.485352 Hz (bin 112.5):
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In pass #4, the frequency of our measured signal is now halfway between two bins, namely between bin 112 and 113. We can see this from the bin magnitude, which reflects this fact by having an identical value for these two bins. Now bin 113 takes on the correct frequency, while bin 112 wraps back to the beginning of its interval (from 3.079 =~ 3.1415 [+?] to -3.1415 [-?]).
For comparison, here’s how the numbers look like when we use an overlap of 75%:
|Pass #5: 2422.485352 Hz (bin 112.5), 4x overlap:
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When we want to alter our signal (which we need to do in order to achieve the pitch shifting effect), we see that with no overlap we easily run into the scenario (depicted in pass #4 above) where on resynthesis we have two sinusoids of equal amplitude but actually one bin apart in frequency. In our case, this would amount to a difference in frequency of about 21.5 Hz. Thus, when putting our signal back together we would have two sinusoids that are apparently 21.5 Hz apart where we had one single sinusoid as input. It is not surprising that the synthesized signal will not sound very much like the original in this case. Clearly, when we just resynthesize the signal without pitch shifting we can expect that these effects will cancel out. However, this no longer holds in the case when we alter our signal by scaling the partial frequencies – we would expect that this will introduce an audible error in our signal since the wrapping will now happen at a different rate.
When we look at pass #5 which uses a 4x overlap (ie. 75%), we see that all adjacent sinusoids down to a bin magnitude of 0.17 (approx. -15 dB) have about the same frequency, the error can be considered negligible. The next sinusoid that does not have the correct frequency is approximately -32dB down, which is much better than in the case where we used no overlap. Thus, we would expect this to sound considerably better. As you can easily verify with the code below, this is indeed the case.
Summarizing the above, we see that choosing a larger overlap, which is actually oversampling our STFT in time, increases our ability to estimate the true frequency of our measured sinusoid signal by making the range in which each sinusoid can deviate from its bin frequency larger: a range of ±1/2 period (±?) has a different meaning when we space our frames more closely, as the same phase difference for 2x and 4x overlap means twice as high a frequency deviation in the 4x case than in the 2x case. Thus, the closer we space our frames, ie. the more they overlap, the better we will be able to determine our true sinusoid frequencies in our measured signal. Of course, the computational cost will also increase, since we need to perform twice as many STFTs when we increase our overlap from 50% to 75%.
As a consequence for its practical implementation we now see why we need an overlap that is sufficiently large: if we have a measured signal whose frequency is between two bins, we will have two bins with large magnitude. However, since the true frequency is somewhere between the two bins and each bin can only deviate by a fixed amount depending on the overlap, we may have two prominent sinusoids that play at two different, yet close frequencies. Even worse, if the true frequency of the sinusoid is between the two bins k and k+1 like in our example shown in pass #4, the frequency of the sinusoid k will move farther away from the true frequency since its phase difference will – trying to lock on the frequency of the input sinusoid – wrap into the opposite branch. This will produce audible beating in our resynthesized sound and thus scramble our result. A 75% overlap remedies this situation: adjacent sinusoids down to an acceptable magnitude are now able to always lock on the true frequency without wrapping – they have become more “flexible” in frequency which will yield a considerably better sonic quality. As a result, the beating is almost completely gone.
5. Shifting the Pitch
Once we have mastered the difficulties of calculating the true partial frequencies for our bins, shifting the pitch is comparably easy. Let’s look at what we now have gained after calculating the partials’ true frequencies: first, we have an array of numbers that holds our magnitude values. When we scale the pitch to become sharp, we expect our magnitude array to expand by our pitch shift factor towards the upper frequency end. Likewise, when we scale our pitch to become flat, we expect our magnitude spectrum to contract towards the low frequency end. Obviously, the actual magnitude values stored in the array elements should not change, as a pure sine of -2dB at 1000 Hz is expected to become a pure sine of -2dB at 500 Hz after a 0.5 x pitch shift. Second, we also have an array that holds the true frequencies of our partial sinusoids. Like with the magnitude spectrum, we would also expect our frequency spectrum to expand or contract according to the scale factor. However, unlike the magnitude values, the values stored in our frequency array denote the true frequencies of our partials. Thus we would expect them to change according to the pitch shift factor as well.
In the simplest of all cases, we can just put the scaled partial frequencies where they belong, and add the magnitudes for the new location together. This will also preserve most of the energy contained in the input.
We also see why pitch shifting using this procedure automatically includes anti-aliasing: we simply do not compute bins that are above our Nyquist frequency by stopping at fftFrameSize2. This does not even need an additional processing step.
6. Back to where we came from…
To obtain our output signal, we need to undo all steps that we took to get our magnitude and frequency spectra. For converting from our magnitude and frequency representation back to bin magnitude, bin frequency and bin phase we follow the same path back that brought us here. Since the sine and cosine parts of the STFT are periodic and defined for all time and not just in a certain interval, we need not care for phase rewrapping explicitely – this is done by the basis functions automatically at no extra cost. After taking the inverse Fourier transform (for which we use the routine smbFft() from our DFT à Pied article) we string our overlapping frames together at the same choice of stride to get back our pitch shifted signal.
7. The Code
So how do we actually implement it in software? First, we obtain our current STFT frame from a FIFO queue. This is required since we don’t want to be forced to use a particular input buffer size – this way we can process all data independent of the FFT frame size we use for the actual pitch shifting. The data is windowed and [re, im] interleaved into the gFFTworksp array where it gets transformed by the Fast Fourier Transform algorithm smbFft(). This FFT algorithm isn’t particularly efficient, it’s just there to demonstrate how to use the routine and to make it compile without modification. You might replace it by your flavor of the FFT later. smbFft() can be found at the end of our DFT à Pied article.
Now we’re equipped with the complex output for the positive and negative frequencies of our DFT bins. Note that we only need the positive frequencies as our original signal was purely real. We convert them to magnitude and phase by rectangular to polar conversion and obtain the instantaneous bin frequency from the phase difference between two adjacent STFT frames. From that we deduce and compensate for the expected, frequency dependent advance in phase between two adjacent frames and obtain the true partial frequency. After doing the actual pitch shifting process described in (6) above, we undo our frequency domain wizardry and then transform to get back to our newly created time domain sequence. After de-interlacing the [re, im] array, windowing and rescaling we put the data into the output queue to make sure we have as much output data as we have input data. The global I/O delay is inFifoLatency samples (which means that the start of your output will contain that many samples of of silence!) – this has to be taken into account when we write the data out to a file.
Worth noting: The routine smbPitchShift() uses static variables to store intermediate results. This is rather inelegant and should be replaced by appropriately allocated and initialized memory instead. Note also on routine parameters: The routine smbPitchShift() takes a pitchShift factor value which is between 0.5 (one octave down) and 2. (one octave up). A value of exactly 1 does not change the pitch. If you need a wider pitch shift range you need to tweak the code a bit. numSampsToProcess tells the routine how many samples in indata[0… numSampsToProcess-1] should be pitch shifted and moved to outdata[0 … numSampsToProcess-1]. This can be any number of samples. The two buffers can be identical (ie. it can process the data in-place). fftFrameSize defines the FFT frame size used for the processing. Typical values are 1024, 2048 and 4096. For a sample rate of 44.1kHz, a value of 1024 is generally fine for speech, while a value of 2048 works well with music. It may be any value <= MAX_FFT_FRAME_LENGTH but it MUST be a power of 2 unless you use an FFT that can deal with other frame sizes as well.
In this article we discussed a way to use the STFT for changing the perceived pitch of an audio signal by representing it as a sum of sinusoids and scaling the frequency of these sinusoids. We have detailed the steps necessary to convert the raw output of a discrete Fourier transform to a musically meaningful representation by making the STFT partials lock and track the actual harmonics in our input signal to the extent permitted by this ‘generic’ representation. We have then used this data to alter the pitch of the underlying signal. C source code for implementing the above process in a black-box type routine that takes and returns any number of sample values as provided by the calling application is given in Appendix A below, subject to the terms set forth in the WOL Wide Open software License.
Stephan M. Bernsee, The DSP Dimension
*) The term “frequency domain” is a common yet somewhat imprecise nickname widely used for the raw output of the Fourier transform. We will use this term in this article only in conjunction with the notion of frequency as introduced by the pitch shifting process implemented below, where it denotes an array of numbers containing the true frequencies of our partials derived by properly post-processing the Fourier transform output.
**) This steps assumes an initial condition for the bin phases which the computation of the sinusoids’ true frequency is based on. In our discussion we will always assume the initial phases are set to zero as the phase difference of the second to the first frame will make successive frequency estimates take on appropriate phase values.
***) Windowing is a process where the signal is faded in and out during one STFT frame in a particular manner. This suppresses the energy that gets smeared over adjacent bins to some extent, since an in such a way tapered signal has less frequencies generated by the discontinuity at the frame boundaries. The interested reader is referred to Chris Bores’ course for a detailed introductory explanation of windowing.