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

794 Chapter 24 The Fourier transform
24.13.3 Parseval’stheorem
If D{f[n]} =F[k] and D{g[n]} =G[k] then
∑N−1
n=0
f[n]g[n] = 1 ∑N−1
F[k]G[k]
N
k=0
where the overline indicates the complex conjugate.
24.13.4 Rayleigh’stheorem
This theorem isobtained fromParseval’s theorem by lettingg[n] = f[n].
∑N−1
|f[n]| 2 = 1 ∑N−1
|F[k]| 2
N
n=0 k=0
Example24.24 VerifyRayleigh’s theoremforthe sequence f[n] = 5,4.
Solution HereN = 2.We first calculateF[k].
F[k] =
1∑
n=0
f[n]e −jnkπ
= f[0] + f[1]e −jkπ
= 5 +4e −jkπ
Whenk =0,wehaveF[0] =5+4e 0 =9.
Whenk=1,wehaveF[1]=5+4e −jπ =5−4=1.
For brevity weoften writeF[k] = 9,1.
Then ∑ 1
n=0 |f[n]|2 =5 2 +4 2 =41.
Also, ∑ 1
k=0 |F[k]|2 = 9 2 +1 2 = 82.
We note that41 = 1 ×82 and this verifies Rayleigh’s theorem.
2
EXERCISES24.13
1 (a)Obtainthe d.f.t.sof f[n] = 1,2,2,1 and
g[n] = 2,1,1,2.
(b)Verify Rayleigh’stheoremforeachofthese
sequences.
2 Suppose f[n] = 3,1,5,4 andg[n] = 2, −1,9,5.
(a) Show that ∑ 3
n=0 f[n]g[n] = 70.
(b) Obtainthe d.f.t.sF[k] andG[k].
(c) CalculateF[k]G[k].
(d) HenceverifyParseval’stheoremforthese
sequences.
3 Show thatifthe d.f.t. of f[n] isF[k] thenthe d.f.t. of
f[n −i] is e −2πjki/N F[k]. This isknown asthe shift
theorem.
4 ProveParseval’stheorem.
5 ProveRayleigh’stheorem.