Signals and Systems - Electrical Engineering

10.4 Discrete Fourier Transform 615

coefficients can be calculated using the Z-transform as

X ̃[k]=^1

N

Z[ ̃x 1 [n]]

∣

∣

z=ejkω^0

=

1

N

N∑− 1

n= 0

̃x[n]e−jω^0 nk 0 ≤k≤N−1, ω 0 = 2 π/N (10.41)

wherex ̃ 1 [n]=x ̃[n]W[n] is a period ofx ̃[n] andW[n] is a rectangular window—that is,

W[n]=u[n]−u[n−N]=

{

1 0≤n≤N− 1

0 otherwise

Thus, the periodic signal ̃x[n] can be expressed as

x ̃[n]=

∑∞

r=−∞

̃x 1 [n+rN] (10.42)

Although one could call Equation (10.41) the DFT of the periodic signal ̃x[n] and Equation (10.40)
the corresponding inverse DFT, traditionally the DFT of ̃x[n] isNX ̃[k], or

X[k]=NX ̃[k]=

N∑− 1

n= 0

x ̃[n]e−jω^0 nk 0 ≤k≤N−1, ω 0 = 2 π/N (10.43)

and the inverse DFT is

̃x[n]=

1

N

N∑− 1

k= 0

X[k]ejω^0 nk 0 ≤n≤N− 1 (10.44)

Equations (10.43) and (10.44) show that the representation of periodic signals is completely discrete:
summations instead of integrals and discrete rather than continuous frequencies. Thus, the DFT and
its inverse can be evaluated by computer.

Given a periodic signalx[n]of periodN, its DFT is given by

X[k]=

N∑− 1

n= 0

x[n]e−j^2 πnk/N 0 ≤k≤N− 1 (10.45)

Its inverse DFT is

x[n]=

1

N

N∑− 1

k= 0

X[k]ej^2 πnk/N 0 ≤n≤N− 1 (10.46)

BothX[k]andx[n]are periodic of the same periodN.