Chapter1:Signals and Signal Processing

Signals play an important role in our daily life

A signal is a function of independent variables such as time,distance,position,temperature, and pressure

A signal carries information

Objective of signal processing:Extract the useful information carried by the signal

Method information extraction:Depends on the type of signal and the nature of the information being carried by the signal

This cource is concerned with the discrete-time representation of signals and their discrete-time processing

Characterization and Classification of signals

Types of signal:Depends on the nature of the independent variables and the value of the function defining the signal

For example, the independent variables can be continuous or discrete

Likewise, the signal can be a continuous or discrete function of the independent variables

Moreover, the signal can be either a real-valued function or a complex-valued function

A signal generated by a single source is called a scalar signal

A signal generated by multiple sources is called a vector signal or a multichannel signal

A one-dimensional (1-D) signal is a funciton of a single independent variable

A multidimensional (M-D) signal is a function of more than one independent variables

The speech signal is an example of a 1-D signal where the independent variable is time

An image signal,such as a photograph, is an exapmle of a 2-D signal where the 2 independent variables are the 2 spatial variables

A color image signal is composed of three 2-D signals representing the three primary colors:red,green and blue(RGB)

A digital signal is thus a quantized sampled-data signal

A continuous-time signal with discrete-value amplitudes is usually called a quantized boxcar signal

The figure illustrates the 4 types of signals

The functional dependence of a signal in its mathematical representation is often explicitly shown

For a continuous-time 1-D signal,the continuous independent variable is usually denoted by t

For example,u(t) represents a continuous-time 1-D signal

For a discrete-time 1-D signal, the discrete independent variable is usually denoted by n

For example,{v[n]} represents a discrete-time 1-D signal

Each member,v[n], of a discrete-time signal is called a sample

In many applications, a discrete-time signal is generated by sampling a parent continuous-time signal at uniform intervals of time

If the discrete instants of time at which a discrete-time signal is defined are uniformly spaced,the independent discrete variable n can be normalized to assume integer value

A signal that can be uniquely determined by a well-defined process,such as a mathematical expression or rule,or table look-up, is called a deterministic signal

A signal that is generated in a random fashion and cannot be predicted ahead of time is called a random signal

Typical Signal Processing Operations

Elementary Time-Domain Operations

Three most basic time-domain signal operations are scaling,delay, and addition

Scaling is simply the multiplication of a signal either by a positive or negative constant

In the case of analog signals, the operation is usually called amplification if the magnitude of the multiplying constant, called gain,is greater than 1

If the magnitude of the multiplying constant is less than 1,the operation is called attenuation

If x(t) is an analog signal that is scaled by a constant $\alpha$ ,then the scaling operation generates a signal $y(t)=\alpha x(t)$

Two other elementary operations are integration and differentiation

The delay operation generates a signal that is a delayed replica of the original signal

For an analog signal x(t),$y(t)=x(t-t_o)$ is the signal obtained by delaying x(t) by the amount of time $t_o$ which is assumed to be a positive number

If $t_o$ is negative,then it is an advance operation

Many applications require operations involving two or more signals to generate a new signal

For example,$y(t)=x_1(t)+x_2(t)+x_3(t)$ is the signal generated by the addition of the three analog signals,$x_1(t),x_2(t),x_3(t)$

The product of 2 signals,$x_1(t),x_2(t)$ generates a signal $y(t)=x_1(t)\ast x_2(t)$

The elementary operations discussed so far are also carried out on discrete-time signals

More complex operations are implemlented by combining two or more elementary operations

Filtering

Filtering is one of the most widely used complex signal processing operations

The system implementing this operation is called a filter

A filter passes certain frequency components without any distortion and blocks other frequency components

Figures below illustrate the lowpass filtering of an input signal composed of 3 sinusoidal components of frequencies 50Hz,110 Hz,and 210 Hz

Figures below illustrate highpass and bandpass filtering of the same input signal



FIR filter

Multiplexing and Demultiplexing

For an efficient utilization of a wideband transmission channel,many narrow-bandwidth low-frequency signals are combined for a composite wideband signal that is transmitted as a single signal

The process of combining the low-frequenct signals is called multiplexing

Multiplexing is implemented to ensure that a replica of each of the original narrow-bandwidth low-frequency signal can be recovered at the receiving end

The recovery process of the low-frequency signals is called demultiplexing

One method of combing different voice signals in a telephone communication system is the frequency-division multiplexing(FDM) scheme

Here,each voice signal, typically bandlimited to a low-frequency band of width ${\Omega }_m$, is frequency-translated into a higher frequency band using the amplitude modulation method

At the receiving end, the composite baseband signal is first recvered from the FDM signal by demodulation

Then each individual frequency-translated signal is demultiplexed by passing the composite signal through a bank of bandpass filters

Reference

1.《Digital Signal Processing：A Computer-Based Approach》