Convolve

Convolve [/A/C] srcWaveName, destWaveName [, destWaveName ]...

The Convolve operation convolves srcWaveName with each destination wave, putting the result of each convolution in the corresponding destination wave.

Convolve is not multidimensional aware. Some multidimensional convolutions are covered by the MatrixConvolve, MatrixFilter, and MatrixOp operations

Flags

/A	Acausal linear convolution.
/C	Circular convolution.

Details

Convolve performs linear convolution unless the /C or /A flag is used. See Convolution for more details.

Depending on the type of convolution, the destination waves' lengths may increase. srcWaveName is not altered unless it also appears as a destination wave.

If srcWaveName is real-valued, each destination wave must be real-valued and if srcWaveName is complex, each destination wave must be complex, too. Double and single precision waves may be freely intermixed; the calculations are performed in double precision.

The linear convolution equation is:

$$\displaystyle {destWaveOut}[p]=\sum_{m=0}^{N-1} {destWaveIn}[m] \cdot {srcWave}[p-m]$$

where N is the number of points in the longer of destWaveIn and srcWave. For circular convolution, the index [p -m ] is wrapped around when it exceeds the range of [0,numpnts(srcWave )-1]. For acausal convolution, when [p -m ] exceeds the range a zero value is substituted for srcWave [p -m ]. Similar operations are applied to destWaveIn [m ].

Another way of looking at this equation is that, for all p, destWaveOut[p] equals the sum of the point-by-point products from 0 to p of the destination wave and an end-to-end reversed copy of the source wave that has been shifted to the right by p.

The following diagram shows the reversed/shifted srcWave that would be combined with destWaveIn. The points numbered 0 through 4 of the reversed srcWave would be multiplied with destWaveIn[0...4] and summed to produce destWaveOut[4]:

For linear and acausal convolution, the destination wave is first zero-padded by one less than the length of the source wave. This prevents the "wrap-around" effect that occurs in circular convolution. The zero-padded points are removed after acausal convolution, and retained after linear convolution. The X scaling of the waves is ignored.

The convolutions are performed by transforming the source and destination waves with the Fast Fourier Transform, multiplying them in the frequency domain, and then inverse-transforming them into the destination wave(s).

The convolution is performed in segments if the resulting wave has more than 256 points and the destination wave has twice as many points as the source wave. For acausal convolution, the length of the resulting wave is considered to be numpnts(srcWaveName)+numpnts(destWaveName)-1 for this calculation.

Applications

The usual application of convolution is to compute the response of a linear system defined by its impulse response to an input signal. srcWaveName would contain the impulse response, and the destination wave would initially contain the input signal. After the Convolve operation has completed, the destination wave contains the output signal.

Use linear convolution when the source wave contains an impulse response (or filter coefficients) where the first point of srcWaveName corresponds to no delay (t = 0).

Use circular convolution for the case where the data in srcWaveName and destWaveName are considered to endlessly repeat (or "wrap around" from the end back to the start), which means no zero padding is needed.

Use acausal convolution when the source wave contains an impulse response where the middle point of srcWaveName corresponds to no delay (t = 0).

See Also

Correlate, Convolution, MatrixOp

References

A very complete explanation of circular and linear convolution can be found in sections 2.23 and 2.24 of Rabiner and Gold, Theory and Application of Digital Signal Processing, Prentice Hall, 1975.