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Lab 3 - Linear Phase, FIR Filter Design by Windowing and IIR Filters

Objectives

FIR filters with linear phase comprise an important class of LTI sytems. This exercise examines the four types of symmetric impulse responses that will result in linear phase. In addition, constraints on the zeros of linear phase filters will be studied and the group delays will be computed. Knowledge of linear phase systems will then be applied to design FIR, linear phase filters by windowing using both parametric and non-parametric windows.

Filter design by windowing involves first calculating the impulse response for an ideal filter. Since the ideal impulse response will not be time-limited, it must be truncated at some point in order to implement it in a practical system. The truncation is done using both non-parametric and parametric windows. Non-parametric windows have a fixed shape and include rectangular, Bartlett and Hamming. With these windows, there is a fundamental tradeoff between main lobe width and side lobe height. Parametric windows, such as the Kaiser window, have shapes that are determined by parameters and can be adjusted to satisfy given constraints on the frequency response.

We will also study FIR filter design by frequency sampling method and Parks-McClellan method. The frequency sampling method allow us to design FIR filters for both typical frequency selective filters (low-pass, high-pass, band-stop and band-pass filters) and filters with arbitrary frequency response. The resulting filter will have a frequency response that is exactly the same as the original response at the sampling instants. However, between the sample points the response will generally be different. To obtain a good approximation to the desired frequency response we must take a sufficient number of frequency samples.

Parks McClellan algorithm is the most widely used algorithm for designing the optimal linear-phase FIR filters. The design is based on the minimization of the peak absolute value (maximum) of the weighted error. The minimized weighted error function exhibits an equi-ripple behavior.

Finally, a filter will be designed using 4 different IIR methods and the results will be compared.

Introduction

The following 4 blocks from J-DSP will be useful in performing this exercise.

The Window Block. J-DSP contains a Window block under the basic blocks menu. This block will be useful in the second and third problems to truncate the ideal impulse response. This block takes an input signal and multiplies it with a window of specified length and type. In the case of the Kaiser window, the parameter must also be entered. After changing any of the settings in the window block, press the update block for the changes to take effect.

The filter blocks menu in J-DSP also contains a Kaiser block and a FIR block. These blocks will produce filter coefficients based on the window design method used in problems 2 and 3, however, they will only give up to 10th order filters. Since the filters in problems 2 and 3 are of orders larger than 10, use the Window block instead.

The Freq. Sampling Block under filter blocks.

The Kaiser Design Block under filter blocks.

The IIR block. The IIR block is also found uder the filter menu and is used for designing IIR filters. To design an IIR filter, specify one of four IIR design methods (Butterworth, Chebychev I, Chebychev II or Elliptic), the filter type(high-pass, low-pass, etc.), the cutoff frequencies and the tolerances. The cutoff frequencies should be entered as percentages of pi. For example, a cutoff frequency of 0.5pi should be entered as 0.5. The IIR block can be connected to a PZ-plot block to see a plot of the filter's poles and zeros and to a Freq-Resp block to see its frequency response. It can also be connected to the bottom of a filter block which will set the filter coefficients of that block to the coefficients produced by the IIR filter design block.

J-DSP

For this lab, use the J-DSP program. Click the link below

Run J-DSP

Problems

Problem 1: FIR linear phase systems

Consider the following four impulse responses.

(a) Find the transfer function for each of the above impulse responses and implement each system as a filter block in J-DSP. Plot the frequency response (magnitude and phase) of each system using linear scaling. Save the graphs of the H₁(jw), H₂(jw), H₃(jw), H₄(jw) as graph1, graph2, graph3 and graph4 respectively.
(b) Derive from the symmetry condition in the time domain (in terms of h[n]), the symmetry condition in the Z-domain (in terms of H(z)).
(c) For each system, describe the symmetries of the zeros of H(z) (use J-DSP's PZ-Plot block to find the roots).
(d) Determine the group delay of each system. Apply a delta at the input of the filter, then take the FFT of the output. Connect the FFT block to a plot block and use the tabulated values in the plot dialog box to derive the exact group delay.(Hint: When calculating the group delay, make sure both vertical and horizontal axis have the same units.)

Problem 2: FIR Filter Design by Windowing

Let

be the ideal impulse response of a low-pass filter. Design a FIR filter with generalized linear phase by truncating this ideal impulse to 60 samples.

For all parts of this problem, use a signal generator block with the following settings:

Signal type: Sinc
Gain: 0.2
Pulsewidth: 120
Periodic: No
Time Shift: 30

These settings provide a shifted, causal version of the impulse response. Use a window block to truncate the ideal impulse response using each of the following window types.

(a) Rectangular
(b) Triangular (Bartlett)
(c) Hamming

(i) Include a diagram of your system in part (a). Save diagram as graph5
(ii) Plot the frequency response of the filter on a dB scale for part (a) and save the plot as graph6.
(iii) Plot the frequency response of the filter on a dB scale for part (b) and save the plot as graph7.
(iv) Plot the frequency response of the filter on a dB scale for part (c) and save it as graph8.
(v) Notice the differences in main lobe width, side lobe height and slope in the transition region of the magnitude response.

Problem 3: Filter design using the Kaiser window method

Design a high-pass filter with generalized linear phase using the Kaiser window method.

Use the following specifications:

(a) Find the minimum length M of the impulse response where M is even. Compute the value of the Kaiser window parameter for a filter that meets the specifications above.
(b) Determine analytically the ideal impulse response.
(c) Use J-Dsp and the Kaiser window determined above to plot the frequency response of the filter on a dB scale and save the plot as graph9. Make sure that your filter has a generalized linear phase.
(d) Find the group delay of the filter.

Hints: To implement this filter, you need two signal generator boxes, an Adder box, a window box,a junction box , FFT box and 2 plot boxes. Subtraction can be done by assigning a negative amplitude to one signal.

Problem 4: Filter design using the frequency sampling block

Step 1:
Design the following low-pass filter. Choose one (1) line segment and number of samples equal to sixteen (16).

Line segments = 1.
N (samples) = 16. (Filter order = 15)

Next we draw the desired (ideal) frequency response that will be sampled at equal intervals. To construct a line segment the user has to place two points by clicking on the desired positions. For a low-pass filter design it is recommended to place the first point on the top left corner with amplitude one and the second point close to the 0.25*pi position (cut-off frequency) with amplitude one.

Place 1^st point à at 0*pi (top left corner) with amplitude one
Place 2^nd point à at 0.25*pi with amplitude one

Press Update to pass the coefficients to the filter. If during the drawing procedure the user wants to correct or start over again, simply press Reset and follow again the instructions above. Observe the frequency response of the designed filter at the output of the FFT block and measure the highest sidelobe level (in dB). Measure also the amplitude level (in dB) at the cut-off frequency 0.25*pi.
Save the following graphs.

The frequency response of the filter (at the output of the FFT block) (graph10)
The drawn frequency response (FreqSamp block) (graph11)

Step 2:

We can improve the amplitude response (decreased sidelobe level) of the frequency sampling filter by introducing a wider transition band between the passband and stopband of the drawn (ideal) frequency response. Design the following low-pass filter. Choose two (2) line segments and number of samples equal to sixteen (16).

Line segments = 2.
N (samples) = 16.

For two line segments the available points to design the desired frequency response are three (3). Place the first point as before (Step 1) and instead of placing the second point at 0.25*pi, place it at 0.2*pi. Finally, put the third point at 0.35*pi with amplitude zero.

Place 1^st point à 0*pi (top left corner) with amplitude one
Place 2^nd point à at 0.2*pi with amplitude one (fourth vertical line)
Place 3^rd point à at 0.35*pi with amplitude zero (seventh vertical line)

Press Update to pass the coefficients to the filter. Observe the frequency response of the designed filter at the output of the FFT block and measure the highest sidelobe level (in dB). Measure also the amplitude level (in dB) at the cut-off frequency 0.25*pi (The cut-off frequency didn’t change due to the wider transition band).
Save the following graphs.

The frequency response of the filters (at the output of the FFT block) (graph12)
The drawn frequency response (FreqSamp block) (graph13)

Problem 5: Filter design using the Parks-McClellan algorithm

Using the Parks-McClellan (PMC) algorithm, design filters with the following specifications.

Filter – 1 specifications

Type – low-pass filter
Cut-off frequencies:
Passband (PB) cut-off frequency: Wp1 = 0.1 * pi
Stopband (SB) cut-off frequency: Ws1 = 0.3 * pi
Tolerances: PB = 3 dB; SB = 20 dB

Design a low-pass filter for the above specifications, and respond to the following:

Save the block diagram as graph14
Save the frequency response in dB as graph15
What is the order of the filter?
Design the FIR filter using the kaiser window method for the above specifications, and find the order of the filter. Compare your results against the PMC method.

Filter - 2 specifications

Type – high-pass filter
Cut-off frequencies:
Passband (PB) cut-off frequency: Wp1 = 0.9 * pi;
Stopband (SB) cut-off frequency: Ws1 = 0.7 * pi
Tolerances:PB = 5 dB; SB = 20 dB.

Design a high-pass filter for the above specifications,

What is the order of the filter?
Design the FIR filter using the kaiser window method for the above specifications, and find the order of the filter. Compare your results against the PMC method.

Problem 6: Filter design comparison using Parks-McClellan, Kaiser and Frequency sampling method

Using the Parks-McClellan, Kaiser (from the FIR design block) and frequency sampling blocks, design filters with the following specifications:

A. Parks-McClellan:

Type: low-pass filter
Cut-off frequencies: Passband (Wp) cutoff frequency: Wp1 = 0.2 * pi.

Stopband (Ws) cutoff frequency: Ws1 = 0.36* pi

Tolerances: PB = 10 dB; SB = 20 dB

B. Kaiser (select Kaiser FIR Filter Design block)

Type: low-pass filter
Cut-off frequencies: Passband (Wp) cutoff frequency: Wp1 = 0.15 * pi.

Stopband (Ws) cutoff frequency: Ws1 = 0.35* pi

Tolerances: PB = 10 dB; SB = 29 dB

C. Frequency sampling

***Same set up as problem 4, step 2.

Observe the frequency response of the designed filters at the output of the FFT block and measure the highest sidelobe level (in dB) for each filter design. Measure also the amplitude level (in dB) at the cut-off frequency 0.25*pi.

Save the following graphs.

The frequency response of the Kaiser Window filter (at the output of the FFT block) (graph16)
The frequency response of the Parks-McClellan filter(at the output of the FFT block) (graph17)
The frequency response of the frequency sampling filter(at the output of the FFT block) (graph18)

Problem 7: IIR Filter Design

In this exercise, you will design an IIR filter with JDSP. The filter will be designed using four different IIR methods (Butterworth, Chebychev I, Chebychev II and Elliptic) so that results of the 4 different methods can be compared. The spefications for the filter are shown below.

Type=Lowpass
Passband cutoff frequency: w_p1=0.4pi
Stopband cutoff frequency: w_s1=0.6pi
Tolerance in passband=1.0dB
Tolerance is stopband= -45.0dB

Use J-DSP's IIR block to design the filter using each one of the four IIR methods mentioned above. You may want to create all 4 of the filters simultaneously with J-DSP so you can compare the frequency response and pole-zero plot of each one. To determine whether the filter is monotonic or equiripple in the stopband, view the frequency response on a dB scale. To determine whether the filter is monotonic or equiripple in the passband, view the frequency response on a linear scale. As you compare the four design methods, answer the following questions.

Which filter requires the highest order to meet the specifications?
Which filter requires the lowest order to meet the specifications?
Which IIR design methods produce a stable filter?
Do any of the four filters have linear phase?
Which of the four filters is equiripple in the passband and monotonic in the stopband?
Which of the four filters is monotonic in the passband and equiripple in the stopband?
Which of the four filters is equiripple in both the stopband and the passband?
Which of the four filters is monotonic in both the stopband and the passband?
For the filters which are monotonic in the passband, where are all the zeros located?

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