Question 24.31: If f[n] is the finite sequence 7. -1 and g[n] is the finite ......
If f[n] is the finite sequence 7. -1 and g[n] is the finite sequence 4, 2, -7 use circular convolution with padded zeros to obtain the linear convolution f * g.
Step-by-Step
Their linear convolution is a sequence of length 2 + 3—1=4.
We pad f and g to give sequences of length 4.
Then, either by direct calculation or by using a computer package you can verify that
F[k]=6,7+\mathrm{j}, 8,7-\mathrm{j} \quad \text { and } \quad G[k]=-1,11-2 \mathrm{j},-5,11+2 \mathrm{j}Then
F[k] G[k]=-6,79-3 \mathrm{j},-40,79+3 \mathrm{j}Then taking the inverse d.f.t. gives
\mathcal{D}^{-1}\{F[k] G[k]\}=28,10,-51,7which is the required linear convolution. You should verify this by calculating the linear convolution directly.
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