Signal Generation in MATLAB: Generating Sine Waves, Square Waves, and Pulse Signals

# 1. Overview of Signal Generation in MATLAB

MATLAB is a powerful computational environment for technical computing that offers a vast library of functions for generating and processing various types of signals. Signal generation is a fundamental task in MATLAB, enabling users to create custom signals for simulation, analysis, and design of a multitude of applications.

This chapter will provide an overview of signal generation in MATLAB, introducing commonly used functions and techniques. It will cover the generation of sine waves, square waves, and pulse signals, and discuss visualization and analysis of signals. By the end of this section, readers will have a solid foundation for generating and processing signals in MATLAB.

# 2. Sine Wave Generation

### 2.1 Mathematical Model of Sine Waves

#### 2.1.1 Definition and Properties of the Sine Function

The sine function is a periodic function, expressed mathematically as:

y = A * sin(2πft + φ)

Where:

- `A` represents the amplitude of the sine wave, indicating the distance between the peaks and troughs.
- `f` is the frequency of the sine wave, indicating how often the peaks or troughs occur within a unit of time.
- `t` is the time variable.
- `φ` is the phase of the sine wave, representing the shift of the waveform along the time axis.

#### 2.1.2 Frequency, Amplitude, and Phase of Sine Waves

The frequency, amplitude, and phase of sine waves are three important parameters that determine the shape and characteristics of the sine wave.

- **Frequency**: The frequency of a sine wave determines the period of the waveform; the higher the frequency, the shorter the period.
- **Amplitude**: The amplitude of a sine wave determines the distance between the peaks and troughs; the greater the amplitude, the more pronounced the peaks and troughs.
- **Phase**: The phase of a sine wave determines the shift of the waveform along the time axis; different phases mean different distances of movement along the time axis.

### 2.2 Generation of Sine Waves in MATLAB

#### 2.2.1 Usage of the sin() Function

The `sin()` function is used to generate sine waves in MATLAB. The syntax of the `sin()` function is as follows:

y = sin(x)

Where:

- `x` is the input angle or radian value.
- `y` is the sine value.

To generate a sine wave, the time variable `t` must be input to the `sin()` function.

#### 2.2.2 Setting Sine Wave Parameters

To generate a sine wave with specific frequency, amplitude, and phase, the corresponding parameters must be set:

- **Frequency**: Use `f = 1/T` to calculate the frequency, where `T` is the period of the sine wave.
- **Amplitude**: Specify the amplitude value directly.
- **Phase**: Use the `φ` parameter to specify the phase shift of the sine wave.

**Code Example:**

% Set sine wave parameters
f = 10; % Frequency at 10 Hz
A = 1; % Amplitude at 1
phi = 0; % Phase shift at 0

% Generate sine wave
t = 0:0.01:1; % Time range
y = A * sin(2*pi*f*t + phi);

% Plot sine wave
plot(t, y);
xlabel('Time (s)');
ylabel('Amplitude');
title('Sine Wave');

**Code Logic Analysis:**

1. Set the sine wave parameters: frequency, amplitude, and phase.
2. Generate the sine wave using the `sin()` function and specify the time range.
3. Plot the sine wave and add labels and a title.

# 3.1 Mathematical Model of Square Waves

#### 3.1.1 Definition and Properties of Square Waves

A square wave is a periodic signal that is not a sine wave, characterized by its square-shaped waveform with two distinct levels. The mathematical model of a square wave can be expressed as:

f(t) = A * sign(sin(2πft + φ))

Where:

- `A` is the amplitude of the square wave
- `f` is the frequency of the square wave
- `φ` is the phase of the square wave
- `sign()` is the sign function, with values as follows:
- When `sin(2πft + φ) > 0`, `sign()` is 1
- When `sin(2πft + φ) < 0`, `sign()` is -1

#### 3.1.2 Duty Cycle and Frequency of Square Waves

The duty cycle of a square wave is defined as the ratio of the duration of the high-level signal to the period. The frequency of a square wave is defined as the number of times a square wave repeats within a period.

**Duty Cycle**

D = (t_high / T) * 100%

Where:

- `t_high` is the duration of the high-level signal
- `T` is the period of the square wave

**Frequency**

f = 1 / T

Where:

- `T` is the period of the square wave

# 4. Pulse Signal Generation

**4.1 Mathematical Model of Pulse Signals**

**4.1.1 Definition and Properties of Pulse Signals**

A pulse signal is an aperiodic signal characterized by a sudden change in amplitude, a short duration, and then a sudden return to its original value. The mathematical model of a pulse signal can be expressed as:

x(t) = A * rect(t/τ)

Where:

- A is the amplitude of the pulse signal
- τ is the width of the pulse signal
- rect(t/τ) is the rectangular function, defined as:

rect(t/τ) = { 1, if |t| ≤ τ/2
               { 0, otherwise

**4.1.2 Amplitude, Width, and Repetition Period of Pulse Signals**

The amplitude, width, and repetition period are three important parameters of pulse signals:

- **Amplitude (A)**: The peak value of the pulse signal.
- **Width (τ)**: The duration of the pulse signal, usually expressed as the full width at half maximum (FWHM), which is the time taken for the pulse signal amplitude to reach half of its peak value.
- **Repetition Period (T)**: For repetitive pulse signals, the repetition period is the interval between two adjacent pulse signals.

**4.2 Generation of Pulse Signals in MATLAB**

**4.2.1 Usage of the pulse() Function**

The MATLAB `pulse()` function is used to generate pulse signals. The syntax of the function is as follows:

y = pulse(t, τ, A, T)

Where:

- y is the output pulse signal
- t is the time vector
- τ is the pulse width
- A is the pulse amplitude
- T is the pulse repetition period

**4.2.2 Setting Pulse Signal Parameters**

The `pulse()` function allows users to set various parameters of pulse signals, including:

***Pulse Width (τ)**: Set the pulse width using the τ parameter.
***Pulse Amplitude (A)**: Set the pulse amplitude using the A parameter.
***Pulse Repetition Period (T)**: Set the pulse repetition period using the T parameter.
***Pulse Shape**: The `pulse()` function supports various pulse shapes, including rectangular, triangular, and Gaussian pulses.

**Code Example**

The following MATLAB code generates a rectangular pulse signal with an amplitude of 1, width of 0.1 seconds, and a repetition period of 1 second:

t = 0:0.001:1; % Time vector
τ = 0.1; % Pulse width
A = 1; % Pulse amplitude
T = 1; % Pulse repetition period

y = pulse(t, τ, A, T);

plot(t, y);
xlabel('Time (seconds)');
ylabel('Amplitude');
title('Rectangular Pulse Signal');

**Code Logic Analysis**

The code first defines the time vector t, then uses the `pulse()` function to generate the rectangular pulse signal y. The input parameters of the `pulse()` function include the time vector t, pulse width τ, pulse amplitude A, and pulse repetition period T.

The `plot()` function is used to draw the pulse signal y. The `xlabel()` and `ylabel()` functions are used to set the labels for the x and y axes. The `title()` function is used to set the title of the graph.

# 5. Signal Generation Practices in MATLAB

### 5.1 Signal Visualization and Analysis

After generating a signal, visualization and analysis are crucial for understanding its characteristics and verifying its correctness. MATLAB provides various functions for signal visualization and analysis.

**plot() Function**

The `plot()` function is used to draw the time-domain waveform of a signal. The syntax is as follows:

plot(t, x)

Where:

* `t`: Time vector
* `x`: Signal data

**Example:**

% Generate a sine wave
t = 0:0.01:1;
x = sin(2*pi*10*t);

% Draw the sine wave
plot(t, x);

**Signal Spectrum Analysis**

Spectrum analysis can reveal the frequency components of a signal. MATLAB provides the `fft()` function for performing a fast Fourier transform, with the syntax as follows:

X = fft(x);

Where:

* `x`: Input signal
* `X`: Spectrum data

**Example:**

% Compute the spectrum of a sine wave
X = fft(x);

% Draw the spectrum
stem(abs(X));

### 5.2 Saving and Loading Signals

For persistent storage and reuse, signals can be saved to a file and loaded when needed. MATLAB provides the `save()` and `load()` functions for saving and loading signals.

**save() and load() Functions**

**save() Function** Syntax:

save('filename.mat', 'x')

Where:

* `filename.mat`: The name of the file to be saved
* `x`: The signal data to be saved

**load() Function** Syntax:

load('filename.mat')

Where:

* `filename.mat`: The name of the file to be loaded

**Example:**

% Save a sine wave
save('sinewave.mat', 'x');

% Load a sine wave
load('sinewave.mat');