Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 87

2.5.1.2 Autocorrelation for Periodic Discrete-Time Power Signals

X

N 1

R x ð k Þ¼ 1
N

x ð n Þ x ð n þ k Þ;

n ¼ 0

where N is the signal period.

2.5.1.3 Energy in Non-Periodic Discrete-Time Energy Signals

R x ð k Þ¼ X

1

T s j x ð n Þj 2

n ¼1

Compare the above formula with the corresponding formula for analog signals:
E ¼ R 1

1 j x ð t Þj 2 dt :

2.5.1.4 Power in Periodic Discrete-Time Power Signals

X

N 1

P ¼ 1
N

j x ð n Þj 2 :

n ¼ 0

2.5.1.5 Parseval's Theorem

N 1

X

j x ð n Þj 2 ¼ X

N 1

For periodic signals: P ¼ 1
N

j X ð k Þj 2
|{z}
PSD

:

n ¼ 0

k ¼ 0

T s j x ð n Þj 2 ¼ Z f s
0

For non-periodic signals: E ¼ X

1

T s j X s ð f Þj 2 df ;

|{z}
ESD

n ¼1

where X s (f) = X(e j2p fT s ) is the DTFT.