[Basics] Signal Spectrum Analysis in MATLAB: Understanding Fourier Transform and Power Spectral Density

# 1. Introduction to Signal Spectrum Analysis

Signal spectrum analysis is a powerful technique used to unveil hidden frequency components within a signal. By decomposing a signal into its individual frequency components, we can gain a deeper understanding of the signal's characteristics, identify patterns, and diagnose problems. Spectrum analysis is applied in many fields, including communications, audio processing, medical imaging, and scientific research.

# 2. Theoretical Foundations of Fourier Transform

### 2.1 Fourier Series and Fourier Transform

**Fourier Series**

The Fourier series is a mathematical tool used to decompose a periodic function into a series of sine and cosine functions. For a periodic function `f(t)` with a period `T`, its Fourier series can be represented as:

f(t) = a_0 + Σ(a_n cos(2πnt/T) + b_n sin(2πnt/T))

where `a_0` is the constant term, `a_n` and `b_n` are Fourier coefficients, which can be calculated using the following formulas:

a_0 = (1/T) ∫[0, T] f(t) dt
a_n = (2/T) ∫[0, T] f(t) cos(2πnt/T) dt
b_n = (2/T) ∫[0, T] f(t) sin(2πnt/T) dt

**Fourier Transform**

The Fourier transform is a mathematical tool that converts a time-domain signal into its frequency-domain representation. For a non-periodic function `f(t)`, its Fourier transform is defined as:

F(ω) = ∫[-∞, ∞] f(t) e^(-iωt) dt

where `ω` denotes the angular frequency.

### 2.2 Properties and Applications of Fourier Transform

The Fourier transform has the following significant properties:

- **Linearity:** `F(af(t) + bg(t)) = aF(f(t)) + bF(g(t))`
- **Time shifting:** `F(f(t - τ)) = e^(-iωτ) F(f(t))`
- **Frequency shifting:** `F(f(t) e^(iω0t)) = F(f(t)) * δ(ω - ω0)`
- **Convolution:** `F(f(t) * g(t)) = F(f(t)) * F(g(t))`

The Fourier transform is widely used in various fields such as signal processing, image processing, and communications, including:

- **Spectrum analysis:** The frequency components of a signal can be obtained by computing the Fourier transform.
- **Signal filtering:** Selective filtering of the Fourier transform can remove specific frequency components from the signal.
- **Image compression:** By decomposing an image into frequency components using the Fourier transform, image compression can be achieved.
- **Communications:** The Fourier transform is used for modulation and demodulation of signals for data transmission.

# 3. Practical Fourier Transform in MATLAB

### 3.1 Using the FFT Function for Fourier Transform

MATLAB provides the `fft` function for performing the Fast Fourier Transform (FFT). The `fft` function takes a