Compensation of undesired phase response in a filter is known as equalization. This is unrelated to the equalization pulses that form part of sync.

The terms nonrecursive and recursive are best used to describe filter implementation structures.

Here I represent frequency by the symbol ω , whose units are radians per second (rad·s-1). A digital filter scales with its sampling frequency; using ω is convenient because the sampling frequency is always ω =2π and the half-sampling (Nyquist) frequency is always π.

Some people define bandwidth differently than I do.

achieve frequency response that might take 20 taps in an FIR filter.

However, there’s a catch: In an IIR filter, both attenuation and delay depend upon frequency. In the terminology of the previous section, an IIR filter exhibits nonlinear phase. Typically, low-frequency signals are delayed more than high-frequency signals. As I have explained, variation of delay as a function of frequency is potentially a very serious problem in video.

An IIR filter cannot have exactly linear phase, although a complex IIR filter can be designed to have arbitrarily small phase error. Because IIR filters usually have poor phase response, they are not ordinarily used in video. (A notable exception is the use of fieldand frame-based IIR filters in temporal noise reduction, where the delay element comprises a field or frame of storage.)

Owing to the dependence of an IIR filter’s result upon its previous results, an IIR filter is necessarily recursive. However, certain recursive filters have finite impulse response, so a recursive filter does not necessarily have infinite impulse response.

Lowpass filter

A lowpass filter lets low frequencies pass undisturbed, but attenuates high frequencies. Figure 20.23 overleaf characterizes a lowpass filter. The response has a passband, where the filter’s response is nearly unity; a transition band, where the response has intermediate values; and a stopband, where the filter’s response is nearly zero. For a lowpass filter, the corner frequency,

ωC – sometimes called bandwidth, or cutoff frequency – is the frequency where the magnitude response of the filter has fallen 3 dB from its magnitude at a reference frequency (usually zero, or DC). In other words, at its corner frequency, the filter’s response has fallen to 0.707 of its response at DC.

The passband is characterized by the passband edge frequency ωP and the passband ripple δP (sometimes denoted δ1). The stopband is characterized by its edge frequency ωS and ripple δS (sometimes denoted δ2).

The transition band lies between ωP and ωS; it has width ∆ω = ωS -ωP.

Insertion gain, relative

1+δp

1

1-δp

0.707 (-3 dB)

+δs

0

-δs

0

∆ω TRANSITION BAND

PASSBAND

CORNER (or CUTOFF, or HALF-POWER) FREQUENCY

STOPBAND

Eq 20.3

Bellanger, Maurice (2000),

Digital Processing of Signals:

Theory and Practice, Third

Edition (Chichester, England:

Wiley): 124.

The complexity of a lowpass filter is roughly determined by its normalized transition bandwidth (or transition ratio) ∆ω/2π. The narrower the transition band, the more complex the filter. Also, the smaller the ripple in either the passband or the stopband, the more complex the filter. FIR filter tap count can be estimated by this formula, due to Bellanger:

In analog filter design, frequency response is generally graphed in log–log coordinates, with the frequency axis in units of log hertz (Hz), and magnitude response in decibels (dB). In digital filter design, frequency is usually graphed linearly from zero to half the sampling frequency. The passband and stopband response of

a digital filter are usually graphed logarithmically; the passband response is often magnified to emphasize small departures from unity.

The templates standardized in BT.601 for a studio digital video presampling filter are shown in

Figure 20.24 opposite. The response of a practical lowpass filter meeting this tremplate is shown in

I describe risetime on page 543. In response to a step input,

a Gaussian filter has a risetime very close to 1⁄3 of the period of one cycle at the corner frequency.

Figure 20.25, on page 215. This is a halfband filter, intended for use with a sampling frequency of 27 MHz; its corner frequency is 0.25fS. A consumer filter might have ripple two orders of magnitude worse than this.

Digital filter design

A simple way to design a digital filter is to use coefficients that comprise an appropriate number of pointsamples of a theoretical impulse response. Coefficients beyond a certain point – the order of the filter – are simply omitted. Equation 20.4 implements a 9-tap filter that approximates a Gaussian:

We could use the term weighting, but sinc itself is a weighting function, so we choose a different word: windowing.

For details about windowing, see Lyons or Rorabaugh, at the end of the chapter, or Wolberg,George (1990), Digital Image Warping

(Los Alamitos, Calif.: IEEE).

Omission of coefficients causes frequency response to depart from the ideal. If the omitted coefficients are much greater than zero, actual frequency response can depart significantly from the ideal.

Another approach to digital filter design starts with the ILPF. Its infinite extent can be addressed by simply truncating the weights – that is, forcing the weights to zero – outside a certain interval, say outside the region 0±4 sample periods. This will have an unfortunate effect on the frequency response, however: The frequency response will exhibit overshoot and undershoot near the transition band.

Poor spectral behavior of a truncated sinc can be mitigated by applying a weighting function that peaks at unity at the center of the filter and diminishes gently to zero at the extremities of the interval. This is referred to as applying a windowing function. Design of a filter using the windowing method begins with scaling of sinc along the time axis to choose the corner frequency and choosing a suitable number of taps. Each tap weight is then computed as a sinc value multiplied by the corresponding window value. A sinc can be truncated through multiplication by a rectangular window. Perhaps the simplest nontrivial window has a triangular shape; this is also called the Bartlett window. The von Hann window (often wrongly called “Hanning”) has

a windowing function that is a single cycle of a raised cosine. Window functions such as von Hann are fixed by the corner frequency and the number of filter taps;

Passband insertion gain [dB]

Stopband insertion gain [dB]

+0.050

+0.025

0

-0.025

-0.050

-10

-20

-30

-40

-50

-60

-70

-80