Figure 20.26 A 25-tap lowpass FIR filter

g[i] = 0.098460si−12 +0.009482si−11 −0.013681si−10 +0.020420si−9

−0.029197si−8

+0.039309si−7

−0.050479si−6

+0.061500si−5

−0.071781si− 4

+0.080612si−3

−0.087404si−2

+0.091742si−1

+0.906788si

+0.091742si+1

−0.087404si+2

+0.080612si+3

−0.071781si+ 4

+0.061500si+5

−0.050479si+6

+0.039309si+7

−0.029197si+8

+0.020420si+9 −0.013681si+10 +0.009482si+11 +0.098460si+12

no control can be exercised over the width of the transition band. The Kaiser window has a single parameter that controls that width. For a given filter order, if the transition band is made narrower, then stopband attenuation is reduced. The Kaiser window parameter allows the designer to determine this tradeoff.

A windowed sinc filter has much better performance than a truncated sinc, and windowed design is so simple that there is no excuse to use sinc without windowing. In most engineering applications, however, filter performance is best characterized in the frequency domain, and the frequency-domain performance of windowed sinc filters is suboptimal: The performance of an n-tap windowed sinc filter can be bettered by an n-tap filter whose design has been suitably optimized.

Few closed-form methods are known to design optimum digital filters. Design of a high-performance filter usually involves successive approximation, optimizing by trading design parameters back and forth between the time and frequency domains. The classic method was published by J.H. McLellan, T.W. Parks, and L.R. Rabiner (“MPR”), based upon an algorithm developed by the Russian mathematician E.Ya. Remez. In the DSP community, the method is often called the “Remez exchange.”

The coefficients of a high-quality lowpass filter for studio video are shown in Figure 20.26 in the margin.

Reconstruction

Digitization involves sampling and quantization; these operations are performed in an analog-to-digital converter (ADC). Whether the signal is quantized then sampled, or sampled then quantized, is relevant only within the ADC: The order of operations is immaterial outside that subsystem. Modern video ADCs quantize first, then sample.

I have explained that filtering is generally required prior to sampling in order to avoid the introduction of aliases. Avoidance of aliasing in the sampled domain has obvious importance. In order to avoid aliasing, an analog presampling filter needs to operate prior to analog-to-digital conversion. If aliasing is avoided, then the sampled signal can, according to Shannon’s theorem, be reconstructed without aliases.

1+ sin0.44 t

2

Figure 20.29 D-to-A conversion with a boxcar waveform is equivalent to a DAC producing an impulse train followed by a boxcar filter with its sinc response. Frequencies close to 0.5fS are attenuated.

1

0.5

0

1

0.5

0

I place “(sin x)/x” in quotes: With the argument properly scaled it is (sin πx)/(πx), but it is almost always pronounced sine-ecks-over-ecks, with argument scaling implicit.

You might consider a DAC’s boxcar waveform to be a “sample-and- hold” operation, but that term is normally used in conjunction with an A-to-D converter, or circuitry that lies in front of an ADC.

“(sin x)/x” correction

I have described how it is necessary for an analog reconstruction filter to follow digital-to-analog conversion. If the DAC produced an impulse “train” where the amplitude of each impulse was modulated by the corresponding code value, a classic lowpass filter would suffice: All would be well if the DAC output resembled my “point” graphs, with power at the sample instants and no power in between. Recall that a waveform comprising just unit impulses has uniform frequency response across the entire spectrum.

Unfortunately for analog reconstruction, a typical DAC does not produce an impulse waveform for each sample. It would be impractical to have a DAC with an impulse response, because signal power is proportional to the integral of the signal, and the amplitude of the impulses would have to be impractically high for the integral of the impulses to achieve adequate signal power. Instead, each converted sample value is held for the entire duration of the sample: A typical DAC produces a boxcar waveform. A boxcar waveform’s frequency response is described by the sinc function.

In Figure 20.29 above, the top graph is a sine wave at 0.44fS; the bottom graph shows the boxcar waveform produced by a conventional DAC. Even with

a high-quality reconstruction filter, whose response extends close to half the sampling rate, it is evident that