Notes on windowed FIR filter design.

Hints video

Some students don't have the textbook and asked me to post the problems. So here they are.

HW 8

1. This problem explores the design a symmetric real-valued linear phase FIR bandpass filter that meets the following specifications.

• Design this filter using the window technique using the best window from the set {rectangular, Bartlett, Hann, Hamming, Blackman}. Indicate which window shape you used.
• Design this filter using the window technique using the Kaiser window. Indicate the Kaiser window parameters ($\beta$$\beta$ and $M$$M$).
• Using FDATool in Matlab, design this filter using equiripple design method.

Make a table with one row for each filter you designed (window, Kaiser, equiripple). The table should have a column for the following parameters: length, order, group delay, type (I, II, III, or IV), and window parameters.

For each filter you design, turn in plots of the impulse response, magnitude response (linear and dB), and phase response. The frequency domain plots should show response over the interval $-\frac{1}{2}\le f\le \frac{1}{2}$$-\frac{1}{2} \leq f \leq \frac{1}{2}$. Also include a pole-zero plot.

2. This problem explores designing a lowpass filter with a very narrow transition band that achieves the following specifications: $A_p=0.1$$A_p = 0.1$ dB ripple in the passband ($0\le |f|\le 0.24$$0 \leq |f| \leq 0.24$) and $A_s=60$$A_s=60$ dB suppression in the stopband ($0.26\le |f|\le 0.5$$0.26 \leq |f| \leq 0.5$).

• Design this filter using the window technique using the best window from the set {rectangular, Bartlett, Hann, Hamming, Blackman}. Indicate which window shape you used.
• Design this filter using the window technique using the Kaiser window. Indicate the Kaiser window parameters ($\beta$$\beta$ and $M$$M$).
• Using FDATool in Matlab, design this filter using equiripple design method.

Make a table with one row for each filter you designed (window, Kaiser, equiripple). The table should have a column for the following parameters: length, order, group delay, type (I, II, III, or IV), and window parameters.

For each filter you design, turn in plots of the impulse response, magnitude response (linear and dB), and phase response. The frequency domain plots should show response over the interval $-\frac{1}{2}\le f\le \frac{1}{2}$$-\frac{1}{2} \leq f \leq \frac{1}{2}$. Also include a pole-zero plot.

3. This problem explores designing a lowpass filter for sample rate conversion using the system depicted in Fig. 6.47 (c) in the textbook (page 379) in which the upsampling factor is $U=3$$U=3$ and the downsampling factor is $D=4$$D=4$. The maximum frequency in the signal $x[n]$$x[n]$ is $f_{\text{max,x}}=\frac{1}{3}$$f_\text{max,x}=\frac{1}{3}$. Design the lowpass filter to have $A_p=0.1$$A_p=0.1$ dB of ripple in the pass band and $A_s=60$$A_s=60$ dB of attenuation in the stopband. Use your knowledge of upsampling and downsampling to determine the passband and stopband edge frequencies, $f_p$$f_p$ and $f_s$$f_s$. What are $f_p$$f_p$ and $f_s$$f_s$?

• Design this filter using the window technique using the best window from the set {rectangular, Bartlett, Hann, Hamming, Blackman}. Indicate which window shape you used.
• Design this filter using the window technique using the Kaiser window. Indicate the Kaiser window parameters ($\beta$$\beta$ and $M$$M$).
• Using FDATool in Matlab, design this filter using equiripple design method.

Make a table with one row for each filter you designed (window, Kaiser, equiripple). The table should have a column for the following parameters: length, order, group delay, type (I, II, III, or IV), and window parameters.

For each filter you design, turn in plots of the impulse response, magnitude response (linear and dB), and phase response. The frequency domain plots should show response over the interval $-\frac{1}{2}\le f\le \frac{1}{2}$$-\frac{1}{2} \leq f \leq \frac{1}{2}$. Also include a pole-zero plot.

4. Suppose you would like to boost by a factor of 4 the low frequencies of an audio signal between the range of $0\le |F|\le 150$$0 \leq |F| \leq 150$ Hz and leave the scale of frequencies above $300$$300$ Hz unchanged. This type of filter has two passbands and no stopband. It is called a shelving filter. The transition band extends from $150$$150$ Hz to $300$$300$ Hz. You plan to perform this processing using the system depicted in Fig. 6.1 of the textbook (page 331) using a sample rate of $\frac{1}{T}=F_s=20$$\frac{1}{T} = F_s = 20$k samples/second. Convert this information to a specification for a discrete-time filter and design a linear phase filter to meet this spec. Require your filter to limit deviations from the desired response to $\delta =0.01$$\delta = 0.01$ in both pass bands.

• Design this filter using the window technique using the best window from the set {rectangular, Bartlett, Hann, Hamming, Blackman}. Indicate which window shape you used.
• Design this filter using the window technique using the Kaiser window. Indicate the Kaiser window parameters ($\beta$$\beta$ and $M$$M$).
• Using FDATool in Matlab, design this filter using equiripple design method.

Make a table with one row for each filter you designed (window, Kaiser, equiripple). The table should have a column for the following parameters: length, order, group delay, type (I, II, III, or IV), and window parameters.

For each filter you design, turn in plots of the impulse response, magnitude response (linear and dB), and phase response. The frequency domain plots should show response over the interval $-\frac{1}{2}\le f\le \frac{1}{2}$$-\frac{1}{2} \leq f \leq \frac{1}{2}$. Also include a pole-zero plot.

5. After working the previous 4 problems, summarize your observations about the best filter design method to use if the goal is to minimize the filter length.

6. Design a differentiator/delay filter pair to be used in FM demodulation. Let the received signal be $x[n]$$x[n]$. The FM demodulated signal is

In FM demodulation (as in many other applications), it is vital to keep signal time aligned. Note that $\text{deriv}\{x[n]\}$$\text{deriv}\{x[n]\}$ is the output of a derivative filter. Because the derivative filter delays the signal, the signal $x[n]$$x[n]$ must also be delayed. Thus it would be more correct to express FM demodulation as

where $\text{delay}\{x[n]\}$$\text{delay}\{x[n]\}$ is the output of an all pass filter having the same delay as the derivative filter.

• Design type III derivative and delay filters using a Hamming window with length $L=21$$L= 21$.
• Design type IV derivative and delay filters using a Hamming window with length $L=22$$L=22$.

In each case, plot the impulse response, and frequency response over $-\frac{1}{2}\le f\le \frac{1}{2}$$-\frac{1}{2} \leq f \leq \frac{1}{2}$.