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

810 Chapter 24 The Fourier transform
Then
F[k]G[k] = −6,79 −3j,−40,79 +3j
Then taking the inverse d.f.t. gives
D −1 {F[k]G[k]} = 28,10,−51,7
whichistherequiredlinearconvolution.Youshouldverifythisbycalculatingthelinear
convolution directly.
Engineeringapplication24.4
Convolutionreverb
Convolutioncanbeusedonanaudiosignaltosimulatetheechoingorreverberation
of a real room or space. An application of such a method can be found in recording
studioswhereitmaybedesirabletosimulatetheacousticsofalargeroomsuchasa
concert hall on a recording.
The first stage involves capturing the impulse response of the room to be simulated.Animpulsivesoundsourcesuchasapistolorasmallexplosioncanbeusedto
exciteabroadrangeoffrequencies.Itcanbeshownthatashort-durationimpulsehas
a very broad spectrum. A popular alternative is to use a sinusoidal signal source of
time-varyingfrequency.Inthelattercasetheoutputoftheconvolutionprocess,h[n],
is known because the spectrum can be measured directly. The second stage consists
ofapplyingtheinverseprocess,termeddeconvolution,tothisdatainordertoobtain
the impulse response of the room.
When the impulse response has been found, whichever method has been used, it
becomes a case of convolving the impulse response of the room or space with the
signal. It is necessary to carry out a linear convolution process to obtain the desired
effect.Wedonotwishtousecircularconvolutionbecausethesignalisnotperiodic.
However, we may wishtomake use of the circular convolution theorem
f ○∗ g =D −1 {D{f[n]} · D{g[n]}}
Recall, however, that this theorem relates only to circular convolution, not to linear
convolution.Wemaystillmakeuseoftheconvolutiontheoremifwepadthesignals
withzerovaluesinordertopreventunwantedoverlap.Thisisbestillustratedbyuse
of anexample.
Theproblemofcalculatingthecircularconvolutionh[n] = f ○∗ gforthesignals
f[n] = 9, −1,3andg[n] = 7,2, −4hasalreadybeenexplained(seeExample24.27).
Theresultofh[n] = 73,−1,−17couldhavebeenobtainedeithergraphicallyorusingtheconvolution
theorem.Thelinearconvolution ish[n] = 63,11, −17,10, −12
and can be found using the direct method, and this result is presented here for reference
(see Exercises 24.15.1, Question 2(c)).
To use the convolution theorem it is first necessary to take the original signals
and pad them with zero values so that the overall length of each is equal to the sum