【Basic】Detailed Explanation of MATLAB Toolbox: DSP System Toolbox

# 2.1 Fundamentals of Digital Signal Processing

Digital Signal Processing (DSP) is the science of processing digital signals, which are signals that vary discretely over time and amplitude. DSP is crucial in modern technology, ranging from audio and image processing to communication and radar systems.

### 2.1.1 Sampling and Quantization

Sampling is the process of converting a continuous-time signal into a discrete-time signal. Quantization is the process of converting a continuous-amplitude signal into a discrete-amplitude signal. Sampling and quantization are fundamental to DSP because they allow us to process continuous signals with digital computers.

### 2.1.2 Fourier Transform

The Fourier Transform is a mathematical tool that converts time-domain signals into frequency-domain signals. The frequency-domain signals show the magnitude and phase of the different frequency components of the signal. The Fourier Transform is widely used in DSP for signal analysis and processing.

### 2.1.3 Filter Design

Filters are circuits or algorithms used to select or remove certain frequency components from a signal. Filters in DSP are used for various applications, such as noise removal, signal enhancement, and feature extraction.

# 2. Theoretical Foundations of the DSP System Toolbox

### 2.1 Fundamentals of Digital Signal Processing

#### 2.1.1 Sampling and Quantization

Sampling is the process of converting a continuous-time signal into a discrete-time signal. It involves obtaining samples from the continuous signal at uniform time intervals. The sampling rate refers to the number of samples per second.

Quantization is the process of converting the continuous amplitude samples into a digital signal with discrete amplitude levels. A quantizer divides the input signal into a finite number of discrete levels and assigns a digital value to each level. Quantization error is the difference between the quantized signal and the original signal.

**Code Block:**

fs = 1000;   % Sampling rate of 1000 Hz
t = 0:1/fs:1;   % Time vector
x = sin(2*pi*100*t);   % Sine wave signal

% Sampling
sampled_x = x(1:10:end);   % Sampling at 10 times the rate

% Quantization
num_levels = 8;   % Number of quantization levels
quantized_x = round(sampled_x * (num_levels - 1)) / (num_levels - 1);

**Logical Analysis:**

* The `fs` variable sets the sampling rate to 1000 Hz.
* The `t` variable creates a time vector, with intervals of 1/fs.
* The `x` variable generates a 100 Hz sine wave signal.
* The `sampled_x` variable samples `x` at 10 times the sampling rate.
* The `num_levels` variable sets the number of quantization levels to 8.
* The `quantized_x` variable quantizes `sampled_x` into 8 levels.

#### 2.1.2 Fourier Transform

The Fourier Transform is a mathematical tool used to convert time-domain signals into frequency-domain signals. It displays the amplitude and phase of different frequency components within the signal.

**Code Block:**

N = 1024;   % FFT length
X = fft(x, N);   % Compute FFT
f = fs * (0:N-1) / N;   % Frequency vector

figure;
plot(f, abs(X));   %