Advanced Chapter: Fundamentals of Signal Processing in MATLAB: Filtering, Fourier Transform, and Spectral Analysis

# 2.1 Filter Types and Characteristics

A filter is a device or algorithm for processing signals, designed to allow certain frequency components to pass through while attenuating others. Depending on the filter's passband and stopband characteristics, they can be classified into the following types:

### 2.1.1 Low-Pass Filter

A low-pass filter permits low-frequency signals to pass while diminishing the amplitude of high-frequency signals. Its passband is from 0 to ωc, and its stopband is from ωc to infinity, where ωc is the cutoff frequency. Low-pass filters are often used to remove high-frequency noise from a signal.

### 2.1.2 High-Pass Filter

A high-pass filter allows high-frequency signals to pass while attenuating low-frequency signals. Its passband is from ωc to infinity, and its stopband is from 0 to ωc. High-pass filters are commonly used to extract the high-frequency components from a signal.

### 2.1.3 Band-Pass Filter

A band-pass filter permits signals within a specific frequency range to pass while attenuating signals at other frequencies. Its passband is from ω1 to ω2, and its stopband is from 0 to ω1 and from ω2 to infinity. Band-pass filters are often used to extract a particular frequency band from a signal.

### 2.1.4 Band-Stop Filter

A band-stop filter permits signals outside a specific frequency range to pass while attenuating signals within that range. Its passband is from 0 to ω1 and from ω2 to infinity, and its stopband is from ω1 to ω2. Band-stop filters are commonly used to remove noise from a particular frequency band within a signal.

# 2. Filter Theory and MATLAB Implementation

### 2.1 Filter Types and Characteristics

A filter is a device or algorithm that selectively allows certain frequency ranges of signals to pass while attenuating or eliminating other frequency ranges. Filters have a wide range of applications in signal processing, such as noise removal, feature extraction, and signal enhancement.

#### 2.1.1 Low-Pass Filter

A low-pass filter allows low-frequency signals to pass, while diminishing the amplitude of high-frequency signals. Its frequency response curve exhibits low-pass characteristics, meaning the amplitude of low-frequency signals remains unchanged, whereas the amplitude of high-frequency signals is attenuated. Low-pass filters are commonly used for removing high-frequency noise and signal smoothing.

#### 2.1.2 High-Pass Filter

A high-pass filter allows high-frequency signals to pass, while attenuating low-frequency signals. Its frequency response curve exhibits high-pass characteristics, meaning the amplitude of high-frequency signals remains unchanged, while the amplitude of low-frequency signals is attenuated. High-pass filters are often used for extracting high-frequency components from a signal and edge detection.

#### 2.1.3 Band-Pass Filter

A band-pass filter allows signals within a specific frequency range to pass, while attenuating signals at other frequencies. Its frequency response curve exhibits band-pass characteristics, meaning the amplitude of signals within the specified passband remains unchanged, whereas the amplitude of signals outside the passband is attenuated. Band-pass filters are commonly used for extracting specific frequency components from a signal.

#### 2.1.4 Band-Stop Filter

A band-stop filter allows signals outside a specific frequency range to pass, while attenuating signals within that range. Its frequency response curve exhibits band-stop characteristics, meaning the amplitude of signals within the specified stopband is attenuated, whereas the amplitude of signals outside the stopband remains unchanged. Band-stop filters are often used for removing interference from a specific frequency in a signal.

### 2.2 Detailed Explanation of MATLAB Filtering Functions

MATLAB offers a variety of filtering functions, ***monly used filtering functions include:

#### 2.2.1 fir1 Function

The `fir1` function is used to design finite impulse response (FIR) filters. Its syntax is:

Hd = fir1(N, Wn, 'type', Window)

Where:

* `N`: The order of the filter
* `Wn`: Normalized cutoff frequency (ranging from 0 to 1)
* `type`: The type of filter, which can be `'low'`, `'high'`, `'bandpass'`, or `'bandstop'`
* `Window`: A window function used to control the filter's frequency response characteristics

**Code Block:**

% Design a low-pass FIR filter with a cutoff frequency of 0.2 and order 20
N = 20;
Wn = 0.2;
Hd = fir1(N, Wn, 'low');

% View the filter's frequency response
freqz(Hd, 1024, 2);

**Logical Analysis:**

This code uses the `fir1` function to design a low-pass FIR filter with an order of 20 and a cutoff frequency of 0.2. The `freqz` function is used to plot the filter's frequency response curve, showing that the filter has good passband characteristics below the cutoff frequency and good stopband characteristics above the cutoff frequency.

#### 2.2.2 butter Function

The `butter` function is used to design Butterworth filters. Its syntax is:

[B, A] = butter(N, Wn, 'type')

Where:

* `N`: The order of