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and denoting

S xx f() = Xf() 2

then the total energy of the signal is given by

E = ∫ ∞ S xx f()df (23.30)
– ∞
where S xx ( f ) represents the distribution of the signal energy as a function of frequency. S xx ( f ) is termed
the energy spectral density for the finite energy signal x(t).
Consider a periodic signal x(t) with an autocorrelation function

1
(
∗
R xx τ() = lim --- ∫ T/2 xt()x t τ) t
–
d
T→∞ T −T/2
Then,
1
R xx 0() = lim --- ∫ T/2 xt() d t (23.31)
2
T→∞ T – T/2
which equals the signal power. Following the same procedure for energy signals, we define S xx ( f ) as the
Fourier transform of R xx (τ) so that

∞
P = R xx 0() = ∫ S xx f() f (23.32)
d
– ∞
and S xx ( f ) is termed the power spectral density of the periodic signal x(t).
The need to analyze stationary random signals also arises in many practical situations. The properties
of such signals can be inferred from their correlation functions. For example the autocorrelation function,
φ xx (τ) of a stationary random signal decreases and goes to zero as τ increases since the events become
uncorrelated for a large separation of time. Hence, φ xx (τ) = φ xx (−τ) and its Fourier transform exists.
Consequently we can write

φ xx 0() = ∫ ∞ Γ xx f() f (23.33)
d
– ∞
where Γ xx ( f ) and φ xx (0) represent, respectively, the power spectral density and the average power of a
random process.

Sampled Continuous-Time Signals

Discrete-time (DT) signals arise either naturally or by sampling continuous-time (CT) signals; however,
the latter form is more often encountered in practice. In this case, a digital signal is formed from a CT
signal through the process of analog-to-digital conversion. The first part of this process is the sampling
of the analog signal, that is, the conversion of x(t) into x(nT s ), where T s is the sampling period and its
reciprocal, F s = 1/T s , is the sampling frequency in samples per second. The sampling frequency must be
appropriately selected to avoid spectral distortion (aliasing), thereby ensuring that x(t) can be reconstructed
from its samples. To gain a good understanding of this procedure the sampling process is examined in the
frequency domain.

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