Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 75

2.3.2 Frequency-Domain Representation of Digital Signals
and Systems

2.3.2.1 Discrete-Time Fourier Series for Periodic Digital Signals

In the analog domain the continuous-time signal and the Fourier Series are related
according to:

Z T o

x ð t Þ¼ X

1

X k e þ j2pkf o t , X k ¼ 1
T o

x ð t Þ e j2pkf o t dt ð 2 : 7 Þ

0

k ¼1

If the analog signal x(t) is sampled with a rate of f s (sampling interval = T s = 1/f s ),
a discrete-time signal x(n) is obtained:

x s ð t Þ¼ x ð n Þ¼ X

1

x ð t ¼ nT s Þ d ð t Þ: ð 2 : 8 Þ

k ¼1

It will be assumed that the period of the above discrete-time signal is N sam-
ples, where N = T o /T s = f s /f o .Ifx s (t) is fed into the formula for the Fourier Series
in Sect. 1.2.3.1 one obtains:

X

N 1

X k ¼ X ð k Þ¼ 1
N

x ð n Þ e j2pkn = N [DFS pair]

n ¼ 0

The above expression is obtained with the result that R T o
0

d ð t Þ dt ¼ 1 : The equation

linking time domain signals with Fourier series for discrete-time signals is

therefore given by:

x ð n Þ¼ X

N 1 N 1

X

X ð k Þ e j2pkn = N , X ð k Þ¼ 1
N

x ð n Þ e j2pkn = N [DFS pair] :

k ¼ 0 n ¼ 0

Note that, unlike the continuous-time FS, the summation for x(n) in the DFS
does not need to go from -? to ? due to the periodicity of both X(k) and the
discrete exponential e j2p kn/N

in k. That is, the DFS coefficients are periodic (with

period N).

2.3.2.2 The Discrete-Time Fourier Transform for Non-Periodic Digital
Signals

Recall from Sect. 2.2 that the formula for the discrete-time Fourier transform
(DTFT) for non-periodic signals is: