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When we're done...

Once a frequency has been found for every window, a vector of frequencies (one for each window) is compiled and returned to the pitch correction handler function.

Waveform of an input audio signal
Waveform of an input audio signal (speech: "Mary had a little lamb...")
Plot of frequency per window
Detected frequencies of the signal above, one per window. Here it is easier to observe the spikes in frequency for parts of speech that may be spoken higher in pitch. If this input was sung rather than spoken, this plot would be much smoother and look closer to the desired frequency.

Desired pitch

The PSOLA pitch correction algorithm requires both an original pitch and a "target" pitch to achieve. If this were a fully-automated pitch-smoothing autotuner, the target pitch would be whatever "note" frequency was closest to the one observed. We on the other hand would like to bend the pitch to the specific frequency of the song, regardless of our starting point. To this end, we must generate a vector of desired frequencies.

Fortunately, thanks to our song interpretation earlier, we already have vectors of the pitch and length of each note in the song at hand. These vectors assume the following format:

  • Frequencies: fundamental frequency in Hz (one per note)
  • Durations: length in seconds independent of sampling frequency (one per note)

First, we generate a vector of frequencies for each sample at our defined sampling rate. This is as simple as producing a vector with a length equal to the total length of the song in seconds times the sampling frequency (thus, lengthN = sum(durations)*Fs). Then, for each note, we copy the frequency of that note over every sample in the vector for a range of the note's duration. This is most easily done using MatLab's "cumsum" function on the durations vector to make each note indexed by the cumulative time passed, and then multiply these by the sampling frequency to produce the index of each note in samples.

Now that we have the frequency for every sample, we can chop up this full-length signal into windows just as we did to the input signal. For each window's range, we simply take the mode of the frequencies in that range (given their short length, a window will never span more than two notes) and let that be the desired frequency for that window.

Plot of the desired pitch per window
A plot of the desired frequency-per-window of "Mary Had a Little Lamb". The high and low notes are very clearly distinguishable.

Psola

Now that we have our original and target frequencies, we can exercise the Pitch-Synchronous Overlap Add algorithm to attempt to correct the frequencies. Like autocorrelation, the PSOLA begins with a windowed, segmented signal. Because we have already determined pitches for a specific number of segments, the PSOLA computations will use the same segment length. This is easy to remember, but introduces some issues. For example, the PSOLA algorithm can make the finest pitch corrections with a greater number of smaller segments, allowing for smoother correction across the signal. But what would happen to the autocorrelation pitch detector if the segment was so small that a full period could not be obtained? A compromise must be made on a segment length which allows for optimal pitch detection and pitch correction, with guesswork as the only means of finding the "happy medium".

Modifying pitch with hanning windows

The signal we input to the PSOLA algorithm is already "windowed" into several overlapping segments. For each segment, the PSOLA creates Hanning windows (windows with a centralized hump-shaped distribution) centralized around the pitch-marks, or spikes in the amplitude. Once the segment is divided into overlapping windows, these windowed areas can be artificially pushed closer together for a shorter, higher-pitched signal, or farther apart for a longer, lower-pitched signal. The jumps between the beginning of each window is shortened or lengthened, and segments are duplicated or omitted where necessary. Unlike resampling, this change of pitch and duration does not compromise the underlying information.

Smoothing it out with overlap and add

Once the pitch and duration of the signal have been adjusted, the segments are then recombined by overlapping and adding. This Overlap-Add method exploits the knowledge that a long discrete convolution can be simplified as the sum of several short convolutions, which is convenient for us since we already have a number of short segments. The Overlap-Add produces a signal which is the same duration as its input and has roughly the same spectrum as the input, but now contains bands of frequency close to our desired frequency and, when played back, shows the result of our desired pitch correction effect.

The PSOLA algorithm described here is the Time-Domain PSOLA. Alternative PSOLA methods exist which depend on linear predictor coefficients rather than segmented waves. The TD-PSOLA is used for its simplicity in programming versus marginal increase in computational cost.

References

Gareth Middleton, "Pitch Detection Algorithms," Connexions, December 17, 2003, (External Link)

Lemmetty, Sami. Review of Speech Synthesis Technology. (Master’s Thesis: Helsinki University of Technology) March 1999. (External Link)

Upperman, Gina. "Changing Pitch with PSOLA for Voice Conversion." Connexions. December 17, 2004. (External Link)

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Source:  OpenStax, Speak and sing. OpenStax CNX. Dec 21, 2009 Download for free at http://cnx.org/content/col11151/1.1
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