DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Strictly speaking, amplitude is an
instantaneous measure that may
take a positive or negative value.
Magnitude is properly either an
absolute value, or a squared or
root mean square (RMS) value
representative of amplitude over
some time interval. The terms are
often used interchangeably.
See Linearity on page 21.
Bracewell, Ronald N., The Fourier
Transform and its Applications,
Second Edition (New York:
McGraw-Hill, 1985).
Magnitude frequency response
To gain a general appreciation of aliasing, it is necessary
to understand signals in the frequency domain. The
previous section gave an example of inadequate
filtering prior to sampling that created an unexpected
alias upon sampling. You can determine whether a filter
has an unexpected response at any frequency by
presenting to the filter a signal that sweeps through all
frequencies, from zero, through low frequencies, to
some high frequency, plotting the response of the filter
as you go. I graphed such a frequency sweep signal at
the top of Figure 7.1, on page 66. The middle graph of
that figure shows a response waveform typical of
a lowpass filter (LPF), which attenuates high frequency
signals. The magnitude response of that filter is shown
in the bottom graph.
Magnitude response is the RMS average response over
all phases of the input signal at each frequency. As you
saw in the previous section, a filter’s response can be
strongly influenced by the phase of the input signal. To
determine response at a particular frequency, you can
test all phases at that frequency. Alternatively, provided
the filter is linear, you can present just two signals –
a cosine wave at the test frequency and a sine wave at
the same frequency. The filter’s magnitude response at
any frequency is the absolute value of the vector sum of
the responses to the sine and the cosine waves.
Analytic and numerical procedures called transforms can
be used to determine frequency response. The Laplace
transform is appropriate for continuous functions, such
as signals in the analog domain. The Fourier transform is
appropriate for signals that are sampled periodically, or
for signals that are themselves periodic. A variant
intended for computation on data that has been
sampled is the discrete Fourier transform (DFT). An
elegant scheme for numerical computation of the DFT
is the fast Fourier transform (FFT). The z-transform is
essentially a generalization of the Fourier transform. All
of these transforms represent mathematical ways to
determine a system’s response to sine waves over a
range of frequencies and phases. The result of a transform
is an expression or graph in terms of frequency.
146 DIGITAL VIDEO AND HDTV ALGORITHMS AND INTERFACES