Chapter 1

Lost Knowledge Refound: Sharpened FIR Filters

Matthew Donadio

Night Kitchen Interactive

What would you do in the following situation? Let’s say you are diagnosing a DSP system problem in the field. You have your trusty laptop with your development system and an emulator. You figure out that there was a problem with the system specifications and a symmetric FIR filter in the software won’t do the job; it needs reduced passband ripple or, maybe, more stopband attenuation. You then realize you don’t have any filter design software on the laptop, and the customer is getting angry. The answer is easy: you can take the existing filter and sharpen it. Simply stated, filter sharpening is a technique for creating a new filter from an old one [1]–[3]. While the technique is almost 30 years old, it is not generally known by DSP engineers nor is it mentioned in most DSP textbooks.

1.1 IMPROVING A DIGITAL FILTER

Before we look at filter sharpening, let’s consider the first solution that comes to mind, filtering the data twice with the existing filter. If the original filter’s transfer function is H(z), then the new transfer function (of the H(z) filter cascaded with itself) is H(z)2. For example, let’s assume the original lowpass N-tap FIR filter, designed using the Parks-McClellan algorithm [4], has the following characteristics:

Number of coefficients: N = 17 Sample rate: Fs = 1 Passband width: fpass = 0.2 Passband deviation: δpass = 0.05 (0.42 dB peak ripple) Stopband frequency: fstop = 0.3 Stopband deviation: δstop = 0.005 (−46 dB attenuation)

Figure 1–1(a) shows the performance of the H(z) and cascaded H(z)2 filters. Everything looks okay. The new filter has the same band edges, and the stopband attenuation is increased. But what about the passband? Let’s zoom in and take a look at Figure 1–1(b). The squared filter, H(z)2, has larger deviations in the passband than the original filter. In general, the squaring process will:

1. Approximately double the error (response ripple) in the passband. 2. Square the errors in the stopband (i.e., double the attenuation in dB in the stopband). 3. Leave the passband and stopband edges unchanged. 4. Approximately double the impulse response length of the original filter. 5. Maintain filter phase linearity.

Figure 1–1 H(z) and H(z)2 performance: (a) full frequency response; (b) passband response.

It is fairly easy to examine this operation to see the observed behavior if we view the relationship between H(z) and H(z)2 in a slightly unconventional way. We can think of filter squaring as a function F[H(z)] operating on the H(z) transfer function. We can then plot the output amplitude of this function, H(z)2, versus the amplitude of the input H(z) to visualize the amplitude change function.

The plot for F[H(z)] = H(z) is simple; the output is the input, so the result is the straight line as shown in Figure 1–2. The function F[H(z)] = H(z)2 is a quadratic curve. When the H(z) input amplitude is near zero, the H(z)2 output amplitude is closer to zero, which means the stopband attenuation is increased with H(z)2. When the H(z) input amplitude is near one, the H(z)2 output band is approximately twice as far away from one, which means the passband ripple is increased.

Figure 1–2 Various F[H(z)] functions operating on H(z).

The squaring process improved the stopband, but degraded the passband. The improvement was a result of the amplitude change function being horizontal at zero. So to improve H(z) in both the passband and stopband, we want the F[H(z)] amplitude function to be horizontal at both H(z) = 0 and H(z) = 1 (in other words, have a first derivative of zero at these points). This results in the output amplitude changing slower than the input amplitude as we move away from zero and one, which lowers the ripple in these areas. The simplest function that meets this will be a cubic of the form

 (1–1)

Its derivative (with respect to x) is

 (1–2)

Specifying F(x) and F′(x) for the two values of x = 0 and x = 1 allows us to solve (1–1) and (1–2) for the cn coefficients as

 (1–3)

 (1–4)

 (1–5)

 (1–6)

Solving (1–5) and (1–6) simultaneously yields c2 = 3 and c3 = –2, giving us the function

 (1–7)

Stating this function as the sharpened filter Hs(z) in terms of H(z), we have

 (1–8)

The function Hs(z) is the dotted curve in Figure 1–2.

1.2 FIR FILTER SHARPENING

Hs(z) is called the “sharpened” version of H(z). If we have a function whose z-transform is H(z), then we can outline the filter sharpening procedure, with the aid of Figure 1–3, as the following:

1. Filter the input signal, x(n), once with H(z). 2. Double the filter output sequence to obtain w(n). 3. Subtract w(n) from 3x(n) to obtain u(n). 4. Filter u(n) twice by H(z) to obtain the output y(n).

Figure 1–3 Filter sharpening process.

Using the sharpening process results in the improved Hs(z) filter performance shown in Figure 1–4, where we see the increased stopband attenuation and reduced passband ripple beyond that afforded by the original H(z) filter.

Figure 1–4 H(z) and Hs(z) performance: (a) full frequency response; (b) passband response.

It’s interesting to notice that Hs(z) has the same half-amplitude frequency (−6 dB point) as H(z). This condition is not peculiar to the specific filter sharpening example used here—it’s true for all Hs(z)s implemented as in Figure 1–3. This characteristic, useful if we’re sharpening a halfband FIR filter, makes sense if we substitute 0.5 for H(z) in (1–8), yielding Hs(z) = 0.5.

1.3 IMPLEMENTATION ISSUES

The filter sharpening procedure is very easy to perform, and is applicable to a broad class of FIR filters; including lowpass, bandpass, and highpass FIR filters having symmetrical coefficients and even-order (an odd number of taps). Even multi-passband FIR filters, under the restriction that all passband gains are equal, can be sharpened.

From an implementation standpoint, to correctly implement the sharpening process in Figure 1–3 we must delay the 3x(n) sequence by the group delay, (N − 1)/2 samples, inherent in H(z). In other words, we must time-align 3x(n) and w(n). This is analogous to the need to delay the real path in a practical Hilbert transformer. Because of this time-alignment constraint, filter sharpening is not applicable to filters having nonconstant group delay, such as minimum phase FIR filters or infinite impulse response (IIR) filters. In addition, filter sharpening is inappropriate for Hilbert transformer, differentiating FIR filters, and filters with shaped bands such as sinc compensated filters and raised cosine filters, because cascading such filters corrupts their fundamental...