DSP Notes: Sampling and Transforms Professor Fred DePiero,
CalPoly State University
Important Points on Sampling 1) The Sampling Theorem states that
a signal can be sampled without loss of information, provided the
sampling rate, Fs (Hz), is at or above the Nyquist rate
Fs >= 2B where B is the bandwidth of the signal. 2) Sampling
a signal x(t) changes its spectrum, X(F). The new spectrum contains
multiple copies of the original version, X(F). See page 272 in the
Proakis text. The copies of the original spectrum appear centered
at multiples of Fs. This replication is due to the fact that
sampling is equivalent to multiplication by an impulse train, d(t),
having impulses separated by 1/Fs (sec). The spectrum of d(t) is
another impulse train, D(F), with impulses separated by Fs (Hz).
Hence the action of multiplying by d(t) in the time domain is
equivalent to convolution with D(F). The convolution operation
results in the spectral copies. 3) When a signal with an analog
frequency F0 is digitized at a sample rate Fs < 2 F0, then
aliasing occurs. In this case, the frequency of the digitized
version of the signal is lower than the original analog version.
For a signal with bandwidth B, as in (1) and (2) above, aliasing
occurs when Fs < 2B and results in overlapping copies of X(F).
Some systems use aliasing in a beneficial way. For example, a
baseband signal can be recovered from a modulated signal through
the process of downconversion, which can be implemented using
aliasing. 4) The relationship between the normalized frequency of a
digitized signal, w0 (radians / sample), and the original analog
frequency, F0 (Hz), is
w0 = 2 pi F0 / Fs This relation is true, irrespective of any
aliasing. An example of a discrete signal at this frequency is x(n)
= cos(w0 n). The samples of x(n) occur at times t = n dT = n / Fs,
dT is the sample spacing (sec). 5) For a given sampling frequency,
Fs, the highest frequency analog signal that can be digitized
without aliasing is at
F0 = Fs / 2 = Fd (Hz) w0 = pi (radians / sample) Analog signals
at higher frequencies cant be appreciated as such in the discrete
domain. When dealing with discrete signals, the most useful range
of the frequency axis is
-Fs / 2
1) Discrete-Time Fourier Transform (DTFT) yields the spectrum
X(w) of a discrete time signal, x(n). X(w) is periodic with period
= 2 pi and the units of w are radians / sample. Typically the range
of: -pi
5) The spectral leakage associated with a finite version of an
infinite duration signal will always be apparent in X(w), as found
via the DTFT. Because the DFT operates on versions of signals
having finite duration, it naturally causes spectral leakage to
occur. However, spectral leakage may not be evident due to the
discrete sampling introduced by X(k). It will not be evident if,
for example, (a) the signal is a single harmonic at F0 where F0 is
an integer multiple of dF, and (b), if the spacing between the
zeros of the sinc function (associated with spectral leakage)
exactly equals the spectral spacing. If there is no zero padding of
the signal then conditions (a) and (b) amount to the requirement
that
F0 N / Fs = c
where c is any integer. So to determine if spectral leakage will
not be observed, compute c, as above. If c is an integer then
leakage will not be observed - regardless of the duration of the
signal, N. 6) Zero padding is the process of concatenating zeros to
the end of a signal. This operation may be performed on any signal,
x(n), prior to computing its transform with a DFT. Zero padding
will cause X(k) to approach X(w). Zero padding helps to reveal
spectral leakage effects. 7) Zero padding has additional uses. In
order to derive the DFT it was necessary to make the assumption
that x(n) is periodic outside the interval of [0, N-1]. This has an
important implication when evaluating a time-domain convolution
x(n)*y(n) by a frequency-domain multiplication of the spectrums
X(k) Y(k), as computed via the DFT. The point of concern arises
because the frequency-domain multiplication corresponds to
convolution of periodic versions of x(n) and y(n). This operation
is referred to as periodic convolution. In general x(n) and y(n)
are not periodic in N. In these cases a technique is needed to
yield the same result as linear convolution would, when evaluated
in the time-domain. If x(n) has length L and y(n) has length M,
then a linear convolution can be achieved by first zero padding
x(n) and y(n) to a length N:
