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 37.

Bracewell, Ronald N. (1985),

The Fourier Transform and Its

Applications, Second Edition (New

York: McGraw-Hill).

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 9.1, on page 98. 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.

Eq 20.1 The sinc function

(pronounced sink) is defined by this equation. Its argument is in radians per second (rad·s-1); here I use the conventional symbol ω for that quantity. The term (sin x)/x (pronounced sine ecks over ecks) is often used synonymously with sinc, without mention of the units of the argument. If applied to frequency in hertz, the function could be written (sin 2πf)/2πf.

sinc is unrelated to sync (synchronization).

Magnitude frequency response of a boxcar

The top graph of Figure 20.7 shows the weighting function of point sampling, as a function of time (in sample intervals). The Fourier transform of the boxcar function – that is, the magnitude frequency response of a boxcar weighting function – takes the shape of

(sin x)/x. The response is graphed at the bottom of Figure 20.7, with the frequency axis in units of

ω =2πfS. Equation 20.1 in the margin defines the function. This function is so important that it has been given the special symbol sinc, introduced by Phillip M. Woodward in 1952 as a contraction of sinus cardinalis.

A presampling filter should have fairly uniform response below half the sample rate, to provide good sharpness, and needs to severely attenuate frequencies at and above half the sample rate, to achieve low aliasing. The bottom graph of Figure 20.7 shows that this requirement is not met by a boxcar weighting function. The graph of sinc predicts frequencies where aliasing can be introduced. Figure 20.6 showed an example of a sine wave at 0.75fS; reading the value of

A near-ideal filter in analog video is sometimes called a brick wall filter, though there is no precise definition of this term.

sinc at 1.5π from Figure 20.7 shows that aliasing is expected.

You can gain an intuitive understanding of the boxcar weighting function by considering that when the input frequency is such that an integer number of cycles lies under the boxcar, the response will be null. But when an integer number of cycles, plus a half-cycle, lies under the weighting function, the response will exhibit a local maximum that can admit an alias.

To obtain a presampling filter that rejects potential aliases, we need to pass low frequencies, up to almost half the sample rate, and reject frequencies above it.

We need a frequency response that is constant at unity up to just below 0.5fS, whereupon it drops to zero. We need a filter function whose frequency response – not time response – resembles a boxcar.

The sinc weighting function

Remarkably, the Fourier transform possesses the mathematical property of being its own inverse (within a scale factor). In Figure 20.7, the Fourier transform of a boxcar weighting function produced a sinc-shaped frequency response. Figure 20.8 opposite shows a sinc-shaped weighting function; it produces a boxcar-shaped frequency response. So, sinc weighting gives the ideal lowpass filter (ILPF), and sinc is the ideal temporal weighting function for use in a presampling filter. However, there are several theoretical and practical difficulties in using sinc. In practice, we approximate it.

An analog filter’s response is a function of frequency on the positive real axis. In analog signal theory, there is no upper bound on frequency. But in a digital filter the response to a test frequency fT is identical to the response at fT offset by any integer multiple of the sampling frequency: The frequency axis “wraps” at multiples of the sampling rate. Sampling theory also dictates “folding” around half the sample rate. Signal components having frequencies at or above the Nyquist rate cannot accurately be represented.

The temporal weighting functions used in video are usually symmetrical; nonetheless, they are usually graphed in a two-sided fashion. The frequency response of a filter suitable for real signals is symmetrical about