This is the second part of a two week laboratory in digital filter design. The first week of the laboratory covered some basic examples of FIR and IIR filters, and then introduced the concepts of filter design. In this week we will cover more systematic methods of designing both FIR and IIR filters.

Download DTFT.m for the following section.

We can generalize the idea of truncation by using different windowing functions to truncate an ideal filter's impulse response. Note that by simply truncating the ideal filter's impulse response, we are actually multiplying (or “windowing”) the impulse response by a shifted rect() function. This particular type of window is called a rectangular window. In general, the impulse reponse h(n) of the designed filter is related to the impulse response h_ideal(n) of the ideal filter by the relation

where w(n) is an N-point window. We assume that

(7.2)

where ω_c is the cutoff frequency and N is the desired window length.

The rectangular window is defined as

(7.3)

The DTFT of w(n) for N=21 is shown in Figure 7.1. The rectangular window is usually not preferred because it leads to the large stopband and passband ripple as shown in Figure 7.2.

Figure 7.1. 

DTFT of a rectangular window of length 21.

Figure 7.2. 

Frequency response of low-pass filter, designed using the truncation method.

More desirable frequency characteristics can be obtained by making a better selection for the window, w(n). In fact, a variety of raised cosine windows are widely used for this purpose. Some popular windows are listed below.

In filter design using different truncation windows, there are two key frequency domain effects that are important to the design: the transition band roll-off, and the passband and stopband ripple (see Figure 7.3 below). There are two corresponding parameters in the spectrum of each type of window that influence these filter parameters. The filter's roll-off is related to the width of center lobe of the window's magnitude spectrum. The ripple is influenced by the ratio of the mainlobe amplitude to the first sidelobe amplitude (or difference if using a dB scale). These two window spectrum parameters are not independent, and you should see a trend as you examine the spectra for different windows. The theoretical values for the mainlobe width and the peak-to-sidelobe amplitude are shown in Table 7.1.

Plot the rectangular, Hamming, Hanning, and Blackman window functions of length 21 on a single figure using the subplot command. You may use the Matlab commands hamming, hann, and blackman. Then compute and plot the DTFT magnitude of each of the four windows. Plot the magnitudes on a decibel scale (i.e., plot ). Download and use the function DTFT.m to compute the DTFT.

Measure the null-to-null mainlobe width (in rad/sample) and the peak-to-sidelobe amplitude (in dB) from the logarithmic magnitude response plot for each window type. The Matlab command zoom is helpful for this. Make a table with these values and the theoretical ones.

Now use a Hamming window to design a lowpass filter h(n) with a cutoff frequency of ω_c=2.0 and length 21. Note: You need to use Equation 7.1 and Equation 7.2 for this design. In the same figure, plot the filter's impulse response, and the magnitude of the filter's DTFT in decibels.

Download nspeech2.mat for the following section.

Figure 7.3. 

Tolerance specifications for the frequency response of a filter.

The standard windows of the "Filter Design Using Standard Windows" section are an improvement over simple truncation, but these windows still do not allow for arbitrary choices of transition bandwidth and ripple. In 1964, James Kaiser derived a family of near-optimal windows that can be used to design filters which meet or exceed any filter specification. The Kaiser window depends on two parameters: the window length N, and a parameter β which controls the shape of the window. Large values of β reduce the window sidelobes and therefore result in reduced passband and stopband ripple. The only restriction in the Kaiser filter design method is that the passband and stopband ripple must be equal in magnitude. Therefore, the Kaiser filter must be designed to meet the smaller of the two ripple constraints:

(7.7)

The Kaiser window function of length N is given by

(7.8)

where I₀(·) is the zero'th order modified Bessel function of the first kind, N is the length of the window, and β is the shape parameter.

Kaiser found that values of β and N could be chosen to meet any set of design parameters, , by defining A=–20log₁₀δ and using the following two equations:

(7.9)

(7.10)

where ⌈·⌉ is the ceiling function, i.e. ⌈x⌉ is the smallest integer which is greater than or equal to x.

To further investigate the Kaiser window, plot the Kaiser windows and their DTFT magnitudes (in dB) for N=21 and the following values of β:

For each case use at least 512 points in the plot of the DTFT.

Figure 7.4. 

DTFT of a section of noisy speech.

Next use a Kaiser window to design a low pass filter, h(n), to remove the noise from the signal in nspeech2.mat using equations Equation 7.1 and Equation 7.2. To do this, use equations Equation 7.9 and Equation 7.10 to compute the values of β and N that will yield the following design specifications:

(7.14)

The low pass filter designed with the Kaiser method will automatically have a cut-off frequency centered between ω_p and ω_s.

(7.15)

Plot the magnitude of the DTFT of h(n) for |ω|<π . Create three plots in the same figure: one that shows the entire frequency response, and ones that zoom in on the passband and stopband ripple, respectively. Mark ω_p, ω_s, δ_p, and δ_s on these plots where appropriate. Note: Since the ripple is measured on a magnitude scale, DO NOT use a decibel scale on this set of plots.

From the Matlab prompt, compute the stopband and passband ripple (do not do this graphically). Record the stopband and passband ripple to three decimal places.

Filter the noisy speech signal in nspeech2.mat using the filter you have designed. Then compute the DTFT of 400 samples of the filtered signal starting at time n=20000 (i.e. 20001:20400 ). Plot the magnitude of the DTFT samples in decibels versus frequency in radians for |ω|<π. Compare this with the spectrum of the noisy speech signal shown in Figure 7.4. Play the noisy and filtered speech signals back using sound and listen to them carefully.

Do the following:

Click here for help on the firpm function for Parks-McClellan filter design. Download the data file nspeech2.mat for the following section.

Kaiser windows are versatile since they allow the design of arbitrary filters which meet specific design constraints. However, filters designed with Kaiser windows still have a number of disadvantages. For example,

Minimizing filter length is important because in many applications the length of the filter determines the amount of computation. For example, an FIR filter of length N may be directly implemented in the time domain by evaluating the expression