Guitar Effects Pitch Scaling
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Time Domain Pitch Scaling using Synchronous Overlap and Add The need for the FFTs can be removed if all of the processing is performed in the time domain using the SOLA approach. This approach works by resampling grains to achieve a change in pitch. In resampling each grain, multirate techniques can be used. Each grain is resampled, and overlapping grains are added together to give the overlap and add method. Figure 21 shows how each grain is processed, with resampling, correlation and windowing.
Figure 21: Block diagram of the overlap and add approach Grain Length and Synchronous Period The grain length (GL, in samples) is determined from the sample rate (FS) used in the system. From experimental results, it was found that for a wide range of guitar signals, that a grain length of between 150 ms and 200 ms maintains a high quality of scaling, with a low level of distortion. Equation 1 shows the relationship with the sample rate for a grain length of 170 ms.
Therefore, at a sampling rate of 24KHz as implemented on the TI DSP, GL = 4080. The Synchronous period (SP) determines the spacing between the start of each grain. This must be considerably smaller than the grain time to ensure that there is adequate overlap between grains. This overlap is required to reduce envelope distortion. The overlap ratio is defined as
Good quality is achieved from RO ³ 0.75, which results in SP = < 1020 samples for the TI DSP implementation. Multirate Signal Processing By performing an upsampling process with interpolation, anti-alias filtering and then downsampling, any desired pitch alteration to the grain can be achieved. Even though the length of the grain is changed, by overlap and adding successive grains the overall length of the signal is maintained. The ratio between semitones is given by
where n is the number of semitone deviations from the original note. Equation (3) allows us to calculate the error in integer ratios when using multirate processing. For example, if the ratio 18:17 is used (upsample by a factor of 18, then downsample by a factor of 17), this gives 1.0588. The exact ratio between one semitone is 1.0595, from (3). This difference of 0.06% is not detectable. However, if we wish to scale up by five semitones using the same upsampling integer (18:23), the error in the frequency increases to 3%, which is detectable. This means that for each semitone increase or decrease in pitch, an integer ratio must be found which has an undetectable error. Table 1 lists the smallest integer ratios that satisfy the requirements, and have been used for implementation. TABLE 1. Integer ratios for multirate processing. Percentage error is error between actual semitone ratio from (3) and integer ratio.
Multirate signal processing is preferred over methods such as FFT because it is less computationally expensive. In addition, the delay in processing data is smaller than FFT as the current grain does not have to be buffered in memory before it is processed. Grain Correlation and Adjustment Once the first part of the grain has been through the resampling process, it is cross-correlated with the corresponding part of the previous grain to find the best position for the grains to be overlapped. This minimises large phase changes, and reduces the effect of waveform distortion through the adding of out of phase grains. Large phase changes at grain boundaries will add unwanted high frequency components. If GL is calculated with equation (1), only a small correlation length is required to obtain high quality results. From testing, this length needs to be approximately 4 ms, or around 100 samples when FS = 24KHz. However, if GL is made to be significantly shorter than that of (1), then cross correlation becomes very important to the quality of the output. The length of the correlation in this case needs to be inversely proportional to GL. The correlation length also depends on the pitch of the signal, with lower frequency notes requiring a longer length to find a good match. Once the maximum point of the cross-correlation function is found, this is used to align the current and previous grains. Grain Windowing Each grain is windowed by a triangular window before grain overlapping and adding. This gives a smooth transition into and out of the grain, and reduces high frequency noise caused from phase changes on grain boundaries when they are added together. The type of window makes little difference to the quality of the output. Other windows such as Hamming and Blackman work just as well. Figure 22 shows the tapering effect of using a triangular window function on a sine wave.
Figure 22: Grain windowing using a triangular function
Figure 23. Comparison between boxcar (rectangular) and triangular grain windowing Figure 23 shows how the phase offsets in the circles in the top subplot are removed by using triangular windowing in the bottom plot. This effect of phase spikes leads to the addition of high frequency noise if not removed. Overlap and Add The next step after windowing is to add the grain to the previous grain with an offset. This offset is the value of the synchronous period, SP plus the offset determined from the cross correlation in section 5.3. This is illustrated by figure 24 on the following page, as the third window of the output signal is the sum of G1C, G2B and G3A. Figure 24 illustrates the entire SOLA process by breaking the input into grains and processing them step-by-step. The areas that are shaded by diagonal lines are where the correlation between grains takes place.
Figure 24: Process of generating the output waveform using SOLA |
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