Question 24.29: Use the convolution theorem to find f ⊛ g when f[n] = 5,4 an......
Use the convolution theorem to find f ⊛ g when f[n] = 5,4 and g[n] = – 1, 3.
Step-by-Step
First we find the corresponding d.f.t.s, F[k] and G[k]. Using
F[k]=\sum_{n=0}^{N-1} f[n] \mathrm{e}^{-2 j n k \pi / N}with N = 2 gives
F[0]=\sum_{n=0}^1 f[n]=9, \quad F[1]=\sum_{n=0}^1 f[n] \mathrm{e}^{-\mathrm{j} n \pi}=5+4 \mathrm{e}^{-\mathrm{j} \pi}=1and similarly,
G[0]=\sum_{n=0}^1 g[n]=2, \quad G[1]=\sum_{n=0}^1 g[n] \mathrm{e}^{-\mathrm{j} n \pi}=-1+3 \mathrm{e}^{-\mathrm{j} \pi}=-4and so
F[k]=9,1 \quad G[k]=2,-4These transforms are multiplied together, term by term, to give
H[k]=F[k] G[k]=(9)(2),(1)(-4)=18,-4
Finally the inverse d.f.t. of the sequence 18, -4, is found using
h[n]=\frac{1}{N} \sum_{k=0}^{N-1} H[k] \mathrm{e}^{2 j n k \pi / N}to give
\begin{aligned} & (f \circledast g)[0]=\frac{1}{2}(18-4)=7 \\ & (f \circledast g)[1]=\frac{1}{2}(18)+\frac{1}{2}\left(-4 \mathrm{e}^{\mathrm{i} \pi}\right)=11 \end{aligned}and so
f \circledast g=7,11The convolution could also be evaluated directly using the technique of Section 24.15.2. You should try this to confirm the result obtained using the theorem.
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