Fundamental of Signals

2-1

Fundamentals of Signals

Practical Signal Processing Concepts and Algorithms using MATLAB

__________________________________________________________________________________________________________

QMS Management Consultants Sdn Bhd (401766-U)

98-1-31 Prima Tanjung Jalan Fettes Tanjung Tokong 11200 Pulau Pinang Malaysia

Telephone +604 899 6020 Facsimile +604 899 4020 E-mail admin@qms-mal.com

11203 FM 2222 #801 Austin Texas 78730

Fundamental of Signals

2-2

What is signal Processing?What is signal Processing?

• The scope of signal processing has grown so broad as to obviate a perfect and precise definition of what is entailed in it

• Traditionally, signal processing includes the materials thought in DSP courses but now signal processing has greater reach because of its influence on related disciplines such as controls, communications theory and also digital communication.

• Thus, signal processing can be defined as that area of applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time, to perform useful operations on those signals.

Fundamental of Signals

2-3

Digital signal processing is the process of extracting useful information from an incoming set of signal (sampled at regular interval, Ts)

When you speak, your voice is picked up by an analog sensor in the cell phone’s microphone

An analog-to-digital converter chip converts your voice (analog) into digital signals, representing 1s and 0s

The Digital Signal Processor (DSP) compresses the digital signals and remove any noise.

In the receiver’s cell phone, a digital-to-analog converter chip changes the digital signal back to an analog voice signal

Your voice exits the phone through the speaker

Fundamental of Signals

2-4

What is Signal Processing Toolbox?What is Signal Processing Toolbox?

• The Signal Processing Toolbox is a collection of tools built on the MATLAB® numeric computing environment.

• The toolbox supports a wide range of signal processing operations, from waveform generation to filter design and implementation, parametric modeling, and spectral analysis.

• The toolbox provides two categories of tools.

Fundamental of Signals

2-5

Section Outline

• Creating and importing signals

• Sampling and re-sampling

• Signal visualization

• Modeling noise

• Modulation

Fundamental of Signals

2-6

Discrete Signals

Time base: t = [0.0 0.1 0.2 0.3];

Signal data: x = [1.0 3.2 2.0 8.5];

Creating vectors in MATLAB:

>> t = [0.0 0.1 0.2 0.3];

>> t = 0:0.1:0.3;

>> t = linspace(0, 0.3, 4);

Fundamental of Signals

2-7

Sampling Signals

Analog signal sources Electromagnetic, audio, sonar, biomedical

Sampling

)()( sa nTxnx

discrete signal

analog signal

sample time

Fundamental of Signals

2-8

Aliasing

Ms ff 2Shannon Sampling Theorem:

Ms ff 2 Ms ff 2

Original signal and sampled signal have same frequency

Sampled signal is aliased to half the original frequency

Fundamental of Signals

2-9

Signal Visualization

• View signal amplitude vs. time index

• Functions plot, stem, stairs, strips

• Listen to data sound

Fundamental of Signals

2-10

Signal Processing Tool

>> sptool

Fundamental of Signals

2-11

Importing a Signal

Choose Import under the File menu

Fundamental of Signals

2-12

• Since MATLAB is a programming language, an endless variety of different signals is possible. Here are some statements that generate several commonly used sequences, including the unit impulse, unit step, and unit ramp functions:

>> t = (0:0.01:1);

>> y = ones(101); % step

>> y = [1; zeros(100,1)]; % impulse

>> y = t ; % ramp

>> y = t.^2; % exponential

>> y = square(2*pi*4*t); % generates a square wave every 0.25secs.

Signal Processing BasicsCommon Sequences

Fundamental of Signals

2-13

Waveform generation

>> y = sin(2*pi*50*t) + 2*sin(2*pi*120*t); %two sinusoids, %one at 50 Hz %and one at %120Hz with %twice the amplitude

>> plot(t,y) %plot y versus time

>> plot(t(1:50),y(1:50)) %display only the first %50 points(zoom!)

Fundamental of Signals

2-14

Signal Browser

Fundamental of Signals

2-15

Changing Sample Rates

To change the sample rate of a signal in SPTool,

1. Click one signal in the signals column of SPTool.2. Select Sampling Frequency in the Edit menu.3. Enter the desired sampling frequency and click OK.

Fundamental of Signals

2-16

Signal GenerationSignal Generation

Signals

• Create a time base vector>> t = [0:0.1:2];

• Signal as function of time>> x = sin(pi*t/2);

Useful MATLAB Functions

• Nonperiodic functionsones, zeros, step

• Periodic functionssin, cos, square, sawtooth

Fundamental of Signals

2-17

Non-periodic Signals

>> t = linspace(0,1,11)

• Step

>> y = ones(11,1);

• Impulse

>> y = [1;zeros(10,1)];

• Ramp

>> y = 2*t;

Fundamental of Signals

2-18

Sine Waves

Parameters

• Amplitude A

• Frequency f

• Phase shift φ

• Vertical offset B

General form

BtfABftAy ))(2sin()2sin(

Fundamental of Signals

2-19

Square Waves

>> sqw1 = square(2*pi*4*t);

>> sqw2 = square(2*pi*4*t,75);

• Duty cycle is 50% (default)• Frequency is 4 Hz

• Duty cycle is 75% • Frequency is 4 Hz

Fundamental of Signals

2-20

Sawtooth Waves

>> saw1 = sawtooth(2*pi*3*t);

>> saw2 = sawtooth(2*pi*3*t,1/2);

• Peak at end of cycle (default)• Frequency is 3 Hz

• Peak halfway through cycle • Frequency is 3 Hz

Fundamental of Signals

2-21

Complex Signals

x(t) = e j2ft = cos(2 ft) + j sin(2 ft) = cos(ωt) + j sin(ωt)

>> x = exp(2*pi*j*f*t);

Useful MATLAB Functions: real, imag, abs, angle

z-plane: e jω Time domain: sin(ωt)

Fs/2 0,Fs

Fundamental of Signals

2-22

textread

xlsreadimread

importdata wavreaduiimport

input

Importing Data

Fundamental of Signals

2-23

Modeling Noise with Random Data

>> un = -5+10*rand(1,1e6);>> hist(un,100)

>> gn = 10+5*randn(1,1e6);>> hist(gn,100)

Uniform Gaussian

Fundamental of Signals

2-24

Adding Noise to a Signal

noisy signal = signal + noise

>> y1 = x + rand(size(x)) uniform noise

>> y2 = x + randn(size(x)) Gaussian noise

Fundamental of Signals

2-25

Pseudorandomness

0.95012928514718

This is the first number produced by the MATLAB uniform random number generator with its default settings. Is it random?

A random sequence is a vague notion ... in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians ...

- D.H. Lehmer

>> rand('state',s) Sets the state to s.

>> rand('state',0) Resets the generator to its initial state.

>> rand('state',sum(100*clock)) New state each time.

Fundamental of Signals

2-26

Resampling

Useful MATLAB functions

downsample, upsample, resample, interp, decimate

Fundamental of Signals

2-27

Modulation and Demodulation

y = modulate(x,fc,fs,'fm')

x = demod(y,fc,fs,'fm')