082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.15 Discrete convolution and correlation 811
of the two − 1. So in the simple example, both signals start with length 3, and after
padding they musthave a length of 3 +3−1 = 5.The padded signals are
f ′ [n] =9,−1,3,0,0
g ′ [n] = 7, 2, −4, 0, 0
The calculation of the first term in the output signal is shown graphically in Figure
24.29.
7
n = 0
0
–1
9
0
2
0
3 0
–4
Figure24.29
The padded signals f ′ [m] (inner circle)andg ′ [m] (outer
circle).
The first element ofh[n] is (9)(7) = 63. Notice how the padding is sufficient to
ensurethatallofthevaluesinthelinearconvolutioncanbeproducedwithoutoverlap.
In this position all of the other terms are multiplied by 0 and hence can be removed
from the calculation. By rotating the outer circle anticlockwise four more times the
other values can be calculated. It can be seen that this method is exactly equivalent
tocarrying out the linear convolution.
Hence wecan usethe convolution theoremon thepadded signals f ′ [n],g ′ [n]and
f ○∗ g= D −1 {D{f ′ [n]} · D{g ′ [n]}}
Thismethodissometimespreferredbecausethed.f.t.valuesofthesignalsarereadily
and quickly calculated on a computer.
The signals may be padded with additional zeros if desired with no effect on the
result. In order to optimize the computation speed for the d.f.t. it is often desirable
to use the f.f.t. algorithm mentioned earlier. One restriction of the f.f.t. is that the
numberofsamplesintheinputsignalmustbeapowerof2,soithastocontain2,4,
8,16,32,... samples.Sincepaddingispossiblewithoutaffectingtheresultthenthe
f.f.t.can be used.
As an example we consider adding reverb to a signalg[n], which is a 30 second
duration piece of music played on an electric guitar. The signalg[n] is very ‘clean’,
obtained bypluggingtheguitardirectlyintotherecording equipment. Asaresult,it
contains no reverberant room effects that would have been present if a microphone
andamplifierhadbeenusedinstead.Animpulseresponse f[n]isobtainedbybursting
a balloon in an environment to be simulated. The two signals prior to introducing
padding areshown inFigure 24.30.
The two padded Fourier transformed signals are multiplied together and the inversed.f.t.takentoproducethesignalinFigure24.31.Thesignalh[n]hasaduration
of 32 s due to the effects of linear convolution. When the signal is played through
speakersithasadistinctiveechoorreverbeffectthatisnotpresenting[n].Thereare
visible differences inthe signalifitis examined carefully.