Frequency Domain Representation in Signals and Systems

# 1. Introduction to Signals and Systems

Signals and systems form the fundamental concepts in the field of signal processing, playing a crucial role in understanding and analyzing various signal characteristics. In this chapter, we will introduce the basic concepts and classifications of signals and systems, as well as the concepts of the time domain and frequency domain.

## 1.1 Concepts and Classifications of Signals

In signal processing, a signal is a description or expression of a phenomenon, which can be either continuous or discrete. Depending on the nature and expression of the signal, it can be divided into various types, such as continuous signals, discrete signals, periodic signals, aperiodic signals, etc.

## 1.2 Concepts and Classifications of Systems

A system is a tool or method for processing or transforming signals. Systems can be linear or nonlinear, time-varying or time-invariant, causal or acausal. Through systems, functions such as signal transmission, filtering, modulation, etc., can be realized.

## 1.3 Concepts of the Time Domain and Frequency Domain

In signals and systems, the time domain refers to the representation of how a signal changes over time, through which we can observe the waveform characteristics of the signal along the time axis. The frequency domain refers to the characteristics of the signal in the frequency domain, through which we can observe the spectral structure and frequency components of the signal. The time domain and frequency domain are mutually corresponding, with the frequency domain representing the information on different frequency components of the signal.

# 2. Fundamentals of Fourier Transform

The Fourier transform is a very important mathematical tool in the field of signals and systems, capable of transforming a signal between the time domain and frequency domain. In this chapter, we will introduce the basic knowledge and related concepts of the Fourier transform.

### 2.1 Fourier Series Representation of Continuous Signals

The Fourier series representation of continuous signals is to express periodic signals as a superposition of a series of sine and cosine functions. Through Fourier series expansion, we can obtain the spectral information of the signal, thus analyzing it in the frequency domain.

import numpy as np
import matplotlib.pyplot as plt

# Generating a triangular wave signal
T = 2*np.pi
t = np.linspace(0, 4*T, 1000)
f = np.zeros_like(t)

for n in range(1, 10, 2):
     f += (4/(n*np.pi))*np.sin(n*t)

plt.plot(t, f)
plt.title('Triangular Wave Signal')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.show()

In the above code, we generated a triangular wave signal and displayed its time-domain waveform.

### 2.2 Fourier Transform of Discrete Signals

The Fourier transform of discrete signals is to represent discrete-time sequences as complex exponential functions in the frequency domain. In digital signal processing, the discrete Fourier transform (DFT) is often used to analyze the spectral characteristics of signals.

import numpy as np

# Generating a discrete signal
x = np.array([1, 2, 3, 4])
X = np.fft.fft(x)

print('Discrete Fourier Transform:')
print(X)

The above code demonstrates how to perform the discrete Fourier transform on a discrete signal using the FFT library function and output the resulting transform.

### 2.3 Properties and Applications of the Fourier Transform

The Fourier transform has properties such as linearity, frequency shift, time shift, etc., which are significant for signal processing and system analysis. For example, the convolution property in the frequency domain can simplify complex operations in signal processing.

Summary: This chapter introduced the basic knowledge of the Fourier transform, including the Fourier series representation of continuous signals, the Fourier transform of discrete signals, and the properties and applications of the Fourier transform. This is important for understanding the analysis and processing of signals in the frequency domain.

# 3. Frequency Domain Analysis

In signals and systems, frequency domain analysis is a vital technique that helps us understand the characteristics and behavior of signals in the frequency domain. Frequency domain analysis mainly involves the spectral density of signals, the frequency response of systems, and the analysis of system characteristics in the frequency domain. We will introduce these contents one by one below:

#### 3.1 Spectral Density of Signals

The spectral density of a signal describes the intensity distribution of the various frequency components of the signal in the frequency domain. In continuous signals, the spectral density can be obtained through the Fourier transform. In discrete signals, we usually use the Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT) to calculate the spectral density. Analysis of spectral density helps us understand the frequency components and energy distribution of the signal.

#### 3.2 Frequency Response of Systems

The frequency response of systems describes how the system responds to signals of different frequencies. In the frequency domain, the system's frequency response is usually represented by a transfer function or frequency response