Signals and systems

EE1205 Introduction to electrical engineeringThe University of Texas at Arlington

Compiled by Md. Raziul HasanRevised and amended by K. Alavi

What is signal?Sign -> Signal• Traffic signal • Flashing indicator light of a car• Mobile network signal• In the fields of communications, signal

processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity.

ref: http://en.wikibooks.org/wiki/Signals_and_Systems/Definition_of_Signals_and_Systems

Signal• "A signal is a function of independent variables

that carry some information.“• e.g. A Voltage V(t) or a Current C(t) that depends

on time

t

V(t)

Complex number representation:

• Three ways to represent a complex number• cartesian z=x+iy• polar• Exponential z=r eiφ

•

z r

2 2r x y and in radians

Complex representation:• Euler's formula: eiφ = cos φ+i sin φThen x= cos φ and y= sin φAnd x=Re {eiφ } and y= Im {eiφ}Let φ= ωt+Then x=Re{ei ωt+}=cos(ωt+)And y=Im{ei ωt+}=sin(ωt+)

y= sin(ωt+) = Im{ei(ωt+)}

How is a Signal Represented?

• Mathematically, signals are represented as a function of one or more independent variables.

• Below signal depends on independent variable t with parameters A, ω and .

• s= A Sin(ωt+)=A Im{ej(ωt+)}

A=Amplitude realω =angular frequency =phaseSin( ~ )=function

ref: http://radarproblems.com/chapters/ch05.dir/ch05pr.dir/c05p1.dir/c05p1.htm

Types of Signal:Analog and digital signals

Continuous and Discrete-Time Signals

Periodic aperiodic Signal

Types of Signal (contd.)• Unit Step Function u(t)

• Ramp function r(t)

System:• A System is any physical set of components that

takes a signal and transforms it into another signal.

• A system can be a simple one that turns a light ON/OFF. Or can be a complex one that does all the computation in a microprocessor.

ref: http://www.dreamstime.com/royalty-free-stock-image-microchip-circuit-image12540796

SystemInput signal x(t)

Output signal y(t)

Amplifier• An electronic amplifier is a device for

increasing the power of a signal.

• It does this by taking energy from a power supply and controlling the output to match the input signal shape but with a larger amplitude.

• There are various types of amplifier.

Amplifier

Time shifter• A time shifter system shifts the function f(t)

forward or backward by a specific time.

• The above system is a forward time shifter. It adds a delay (t0) to the signal.

t tTime

shifter

f(t) f(t – t0)

Sampler• sampling is the reduction of a continuous-time

signal to a discrete-time signal

• The sampling frequency must be higher than the frequency of the signal to be sampled. (minimum twice as high)

Sampler

ref: http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29

Analog to Digital Converter• An analog-to-digital converter (ADC, A/D) is a

device that converts a continuous quantity to a discrete time digital representation.

• The system that does the opposite is called DAC

Analog to Digital

ref: http://pictureofgoodelectroniccircuit.blogspot.com/2010/04/phase-and-function-of-analog-signal-or.html

DAC

Analog to Digital Conversion

• Conversion of Analog to digital is done in two step.

1. Continuous analog Sampled signal2. Sampled signal Quantized digital signal

A/DSampler

ref: http://en.wikipedia.org/wiki/Quantization_%28signal_processing%29

Low Pass filter• A low-pass filter is a filter that passes low-

frequency signals but attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency.

ref: http://en.wikipedia.org/wiki/Low-pass_filter

Low Pass Filter

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High Pass filter• A high-pass filter (HPF) is a device that

passes high frequencies and attenuates (i.e., reduces the amplitude of) frequencies lower than its cutoff frequency.

ref: http://en.wikipedia.org/wiki/High-pass_filter

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Band Pass filter• A band-pass filter is a device that passes

frequencies within a certain range and rejects (attenuates) frequencies outside that range.

ref: http://en.wikipedia.org/wiki/Band-pass_filter

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Band Stop filter• band-stop filter or band-rejection filter is a

filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels

ref: http://en.wikipedia.org/wiki/Band-stop_filter

Band –stop Filter

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Signal Processing:• In signal processing, sampling

is the reduction of a continuous signal to a discrete signal.

• Sampling frequency defines the sampling ‘RESOLUTION’

ref: http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29, http://fourier.eng.hmc.edu/e101/lectures/Sampling_theorem/node1.html

Signal processing:• Fourier Series expansion:

ref: http://mathworld.wolfram.com/FourierSeries.html

The first four Fourier series approximations

for a square wave.

The Fourier Series expansion:• A periodic square

wave function ƒ(x) can be expressed with infinite Fourier series expansion.

f(x)=sin(x)+ sin(3x)+sin(5x)+sin(7x)+…..

ref: http://www.intmath.com/fourier-series/2-full-range-fourier-series.php

Figure: Construction of a square wave from sinusoids of different frequencies

Application:• Communication Applications:

Transmission of information (signal) over a channel “Modulation”

• Control Applications:Control the speed of fan

• Signal Processing Applications:o Speech and audio processingo Multimedia processing

• Biological Signal Analysis:o Brain signals (EEG)o Cardiac signals (ECG)o Medical images (x-ray, PET, MRI)

ref: http://images.yourdictionary.com/amplitude-modulation, http://en.wikipedia.org/wiki/Amplitude_modulation

Amplitude Modulator• An Amplitude modulator takes a low frequency

message signal and a high frequency carrier signal and combines it into a modulated signal of varying amplitude.

ref: http://encyclopedia2.thefreedictionary.com/Amplitude+Modulation

Amplitude Modulator

Amplitude Demodulator• An Amplitude demodulator takes a modulated

signal as input and gives output the original message signal.

• The above system is an amplitude demodulator. Its also called ‘envelope detector’. It takes the envelope of the modulated signal to get the information signal.

ref: http://encyclopedia2.thefreedictionary.com/Amplitude+Modulation

Amplitude Demodulator