FIR Filter Design using the Z-Transform

A small MATLAB project that designs two first-order FIR filters — a low-pass and a high-pass — starting from their Z-domain transfer functions, and verifies the design three different ways: frequency response, pole-zero map, and impulse response.

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Contents

• Why
• How it works
• Results
• Running it
• Repository structure
• What I'd add next
• License

Why

Most intro DSP courses teach you to write a difference equation and just trust that it filters the way it's supposed to. This project goes the other way: start from H(z), derive the difference equation algebraically using MATLAB's Symbolic Math Toolbox, and then check the result three separate ways instead of just eyeballing one plot.

Filter	H(z)	Difference equation
Low-Pass	(z + 1) / z	y[n] = x[n] + x[n-1]
High-Pass	(z - 1) / z	y[n] = x[n] - x[n-1]

The low-pass filter averages two consecutive samples, so fast changes get smoothed out while slower trends survive. The high-pass filter subtracts consecutive samples instead, so it does the opposite — it kills slow-moving/DC content and keeps the sharp, fast-changing parts. It's essentially a first-order discrete derivative, which is why it's useful for things like edge detection or removing a DC offset from a signal.

Both are also a good sanity check for the "FIR" part of the name: a system is only truly FIR if its impulse response actually dies out after a finite number of samples, rather than that just being assumed.

How it works

1. Define H(z) symbolically for each filter using syms.
2. Pull out the numerator and denominator polynomials with numden, then convert them to plain coefficient vectors using sym2poly — this is the form freqz, zplane, and impz all expect.
3. Find the poles and zeros with roots, and check stability by confirming every pole lies inside the unit circle.
4. Plot three views of the same system for each filter, and save all of them to images/:
   • Frequency response (freqz) — magnitude in dB and phase in degrees, 0 to π
   • Pole-zero map (zplane) — where the poles/zeros actually sit relative to the unit circle
   • Impulse response (impz) — confirms the response is finite, which is the entire point of "FIR"

Both filters are handled by the same analyze_filter function in the script, so there's no copy-pasted logic between the LPF and HPF sections — just two calls with a different H(z) each.

Results

High-pass — attenuates near DC, opens up as frequency increases toward Nyquist. The pole-zero map shows the zero sitting right on the unit circle at z = 1 (DC), which is exactly why DC gets killed.

Frequency response	Pole-zero map	Impulse response

Low-pass — the mirror image: strong near DC, rolls off toward Nyquist. Here the zero sits at z = -1 (Nyquist) instead.

Frequency response	Pole-zero map	Impulse response

The two frequency responses are complementary, which is exactly what you'd expect from (z+1)/z and (z-1)/z. Both impulse responses are non-zero for only 2 samples, confirming both systems are genuinely FIR — there's no pole anywhere except at the origin, so nothing "rings" or decays indefinitely.

Running it

You'll need MATLAB with:

• Symbolic Math Toolbox — for the syms/numden/sym2poly steps
• Signal Processing Toolbox — for freqz, zplane, and impz

fir_filter_design

This prints the coefficients, zeros, poles, and a stability check for both filters to the console, opens three figures per filter, and (re)saves the corresponding PNGs into images/.

Repository structure

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├── fir_filter_design.m   # Main script — defines both filters and calls analyze_filter
├── images/
│   ├── low_pass_frequency_response.png
│   ├── low_pass_pole_zero_map.png
│   ├── low_pass_impulse_response.png
│   ├── high_pass_frequency_response.png
│   ├── high_pass_pole_zero_map.png
│   ├── high_pass_impulse_response.png
│   └── filter_response.fig   # Editable MATLAB figure
└── README.md

What I'd add next

• Generalize analyze_filter to accept arbitrary-order H(z) instead of hardcoding two first-order cases
• Run both filters on an actual noisy test signal (not just the theoretical response) to show the filtering effect directly
• Compare against fir1/designfilt to see how a "designed from scratch" filter stacks up against MATLAB's built-in filter design tools

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Licensed under MIT — see LICENSE.