Summary of Digital Signal Processing with MATLAB and Python

Digital Signal Processing with MATLAB and Python Guide

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Introduction

Digital filter design is the process of creating discrete-time systems that shape or modify signals in frequency and time. This material covers IIR and FIR design methods, practical implementation in MATLAB/Python, analysis tools (frequency, impulse, step responses), and common workflows used in engineering practice.

Definition: A digital filter is a discrete-time system that maps an input sequence $x[n]$ to an output sequence $y[n]$ via a linear time-invariant relationship realized by difference equations or convolution with an impulse response.

1. Key concepts and building blocks

1.1 Analog prototypes and bilinear transform

Many IIR digital filters are created by transforming an analog prototype $F(s)$ into a digital transfer function $G(z)$ using the bilinear transform.
The bilinear transform maps the $s$-plane to the $z$-plane by substituting $$s = \frac{2F_s\left(1 - z^{-1}\right)}{1 + z^{-1}}$$
Given prototype coefficient vectors $B_s$ and $A_s$, toolboxes provide a function to convert them: in Python SciPy:
```
from scipy.signal import bilinear
B_z, A_z = bilinear(B_s, A_s, Fs)
```

Definition: The bilinear transform is a conformal map that preserves stability and maps the imaginary $s$-axis to the unit circle in the $z$-plane, avoiding aliasing of frequencies but introducing frequency warping.

1.2 Second-order (biquad) sections and quality factor $Q$

Second-order low-pass prototype commonly written as $$F(s)=\frac{\omega_c^2}{s^2 + \frac{\omega_c}{Q}s + \omega_c^2}.$$
Parameters:
- $\omega_c$ — cutoff (rad/s)
- $Q$ — quality factor controlling peak/damping
Poles location determines time-domain behavior (overdamped, critically damped, underdamped, unstable).

💡 Věděli jste?Fun fact: Many practical IIR filters are implemented as cascades of second-order sections (biquads) because numerical stability and coefficient quantization are easier to manage than high-order single-polynomial implementations.

1.3 IIR vs FIR trade-offs (comparison)

Property	IIR	FIR
Stability guarantee	Needs pole check (can be unstable if designed improperly)	Always stable if causal finite-length impulse response
Phase	Non-linear typically	Can be linear (with symmetric coefficients)
Complexity for sharp response	Lower order often	Higher order needed
Implementation	Requires feedback (difference eq.)	Pure convolution (no feedback)

1.4 FIR design methods covered

Window method (firwin): design by truncating ideal impulse response and applying a window (Hamming, Hann, Blackman, Kaiser, ...)
Least-squares method (firls): minimize squared error across bands, define edges and target gains explicitly

1.5 IIR design methods covered

Direct design from specifications (iirdesign, iirfilter) using elliptic, Butterworth, Chebyshev types
Design parameters: passband edge $f_{p}$, stopband edge $f_{s}$, passband ripple $A_{p}$ (dB), stopband attenuation $A_{s}$ (dB)

Definition: Passband ripple $A_{p}$ is the maximum allowed deviation (in dB) from 0 dB within the passband; stopband attenuation $A_{s}$ is the minimum required attenuation (in dB) in the stopband.

2. Practical tools and functions (MATLAB/Python)

2.1 Frequency response analysis

Analog: MATLAB/Python function freqs for prototype analog transfer functions.
- Example usage in Python:
```
from scipy.signal import freqs
w, K = freqs(B, A, w)
```
Digital: freqz computes discrete-time frequency response from numerator/denominator or SOS.
- Example:
```
from scipy.signal import freqz
W, H = freqz(B, A)
f = W/(2*pi)*Fs
```

2.2 Time-domain responses

Unit impulse response: for FIR it is the filter coefficients; for IIR it can be obtained by filtering an impulse.
Unit step response: useful for transient behavior and control applications (e.g., suspension design).
Compute using filter (MATLAB) or lfi

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Digital Filter Design

Klíčové pojmy: Use bilinear transform to convert analog prototype coefficients using $s=\frac{2F_s(1-z^{-1})}{1+z^{-1}}$, Second-order prototype: $F(s)=\frac{\omega_c^2}{s^2+\frac{\omega_c}{Q}s+\omega_c^2}$ and vary $Q$ to change damping, In Python: bilinear(B_s,A_s,Fs) returns digital numerator and denominator, Use freqz for digital and freqs for analog frequency responses, Design IIR with iirdesign(iirfilter) specifying $f_{p}$, $f_{s}$, $A_p$, $A_s$, and fs, FIR via firwin (window) or firls (least-squares) — FIR has linear phase, Implement high-order IIR as cascade of SOS (biquads) to improve numeric stability, Validate designs with magnitude, phase, pole-zero, impulse, and step responses, For offline filtering use lfilter (SciPy) or filter (MATLAB), Pre-warp frequencies when using bilinear transform or use library options that handle warping, Check pole locations to ensure digital filter stability, Pass explicit fs in SciPy functions to avoid normalized-frequency mistakes

## Introduction Digital filter design is the process of creating discrete-time systems that shape or modify signals in frequency and time. This material covers IIR and FIR design methods, practical implementation in MATLAB/Python, analysis tools (frequency, impulse, step responses), and common workflows used in engineering practice. > Definition: A digital filter is a discrete-time system that maps an input sequence $x[n]$ to an output sequence $y[n]$ via a linear time-invariant relationship realized by difference equations or convolution with an impulse response. ## 1. Key concepts and building blocks ### 1.1 Analog prototypes and bilinear transform - Many IIR digital filters are created by transforming an **analog prototype** $F(s)$ into a digital transfer function $G(z)$ using the **bilinear transform**. - The bilinear transform maps the $s$-plane to the $z$-plane by substituting $$s = \frac{2F_s\left(1 - z^{-1}\right)}{1 + z^{-1}}$$ - Given prototype coefficient vectors $B_s$ and $A_s$, toolboxes provide a function to convert them: in Python SciPy: ```python from scipy.signal import bilinear B_z, A_z = bilinear(B_s, A_s, Fs) ``` > Definition: The bilinear transform is a conformal map that preserves stability and maps the imaginary $s$-axis to the unit circle in the $z$-plane, avoiding aliasing of frequencies but introducing frequency warping. ### 1.2 Second-order (biquad) sections and quality factor $Q$ - Second-order low-pass prototype commonly written as $$F(s)=\frac{\omega_c^2}{s^2 + \frac{\omega_c}{Q}s + \omega_c^2}.$$ - Parameters: - $\omega_c$ — cutoff (rad/s) - $Q$ — quality factor controlling peak/damping - Poles location determines time-domain behavior (overdamped, critically damped, underdamped, unstable). Fun fact: Many practical IIR filters are implemented as cascades of second-order sections (biquads) because numerical stability and coefficient quantization are easier to manage than high-order single-polynomial implementations. ### 1.3 IIR vs FIR trade-offs (comparison) | Property | IIR | FIR | |---|---:|---:| | Stability guarantee | Needs pole check (can be unstable if designed improperly) | Always stable if causal finite-length impulse response | | Phase | Non-linear typically | Can be linear (with symmetric coefficients) | | Complexity for sharp response | Lower order often | Higher order needed | | Implementation | Requires feedback (difference eq.) | Pure convolution (no feedback) | ### 1.4 FIR design methods covered - Window method (firwin): design by truncating ideal impulse response and applying a window (Hamming, Hann, Blackman, Kaiser, ...) - Least-squares method (firls): minimize squared error across bands, define edges and target gains explicitly ### 1.5 IIR design methods covered - Direct design from specifications (iirdesign, iirfilter) using elliptic, Butterworth, Chebyshev types - Design parameters: passband edge $f_{p}$, stopband edge $f_{s}$, passband ripple $A_{p}$ (dB), stopband attenuation $A_{s}$ (dB) > Definition: Passband ripple $A_{p}$ is the maximum allowed deviation (in dB) from 0 dB within the passband; stopband attenuation $A_{s}$ is the minimum required attenuation (in dB) in the stopband. ## 2. Practical tools and functions (MATLAB/Python) ### 2.1 Frequency response analysis - Analog: MATLAB/Python function freqs for prototype analog transfer functions. - Example usage in Python: ```python from scipy.signal import freqs w, K = freqs(B, A, w) ``` - Digital: freqz computes discrete-time frequency response from numerator/denominator or SOS. - Example: ```python from scipy.signal import freqz W, H = freqz(B, A) f = W/(2*pi)*Fs ``` ### 2.2 Time-domain responses - Unit impulse response: for FIR it is the filter coefficients; for IIR it can be obtained by filtering an impulse. - Unit step response: useful for transient behavior and control applications (e.g., suspension design). - Compute using filter (MATLAB) or lfi