CEG 383/EEE221/ETE221: Signals and SystemsLec 06: Fourier and Laplace TransformFaculty: Dr. M. RokonuzzamanCell: zaman.rokon@yahoo.com

OutlineTransforms in context of problem solvingConvolutionDirac delta function d(t)Fourier TransformSamplingLaplace Transform

Why use Transforms?Transforms are not simply math curiosity sketched at the corner of a woodstove by ol Frenchmen.Way to reframe a problem in a way that makes it easier to understand, analyze and solve.

General Scheme using TransformsProblemEquationof the problemSolutionof the equationResultTransformationInversetransformationTransformedequationSolution of thetransformed equation= HARD= EASY

Which Transform to Use?

ApplicationContinuousDomainDiscreteDomainSignal ProcessingFourier T.Discrete F.T. (DFT/FFT)Control TheoryLaplace T.z-Transform

Typical ProblemGiven an input signal x(t), what is the output signal y(t) after going through the system?To solve it in the time domain (t) is cumbersome!

System/Filter

t

x(t)

Integrating Differential Equation?Lets have a simple first order low-pass filter with resistor R and capacitor C:

The system is described by diff. eq.:

To find a solution, we can integrate. Ugh!

ConvolutionMath operator (symbol *) that takes two input functions (x(t) and h(t)) and produces a third (y(t))

Expresses the amount of overlap of one function x(t) as it is shifted over another function h(t).

Way of blending one function with another.

Fourier TransformJean-Baptiste Fourier had crazy idea (1807):

Any periodic function can be rewritten as a weighted sum of sines and cosines of different frequencies.

Called Fourier Series

Square-Wave Deconstruction

Other examples

FT expands this ideaTake any signal (periodic and non-periodic) in time domain and decompose it in sines + cosines to have a representation in the frequency domain.

t

FT: Formal DefinitionConvention: Upper-case to describe transformed variables:Transform: F{ x(t) } = X(w) or X(f) (w=2pf)Inverse: F-1{Y(w) or Y(f) }= y(t)

FT gives complex numbersYou get complex numbersCosine coefficients are realSine coefficients are imaginary

t

Complex planeComplex number can be represented:Combination of real + imaginary value:x +iy

Amplitude + PhaseA and j

Alternative representation of FTComplex numbers can be represented also as amplitude + phase.

t

Example Fourier TransformFast moving vs slow moving signals

FT

f

AmplitudeSpectrum

t

Example Fourier Transform Time Domain t Frequency Domain wRealRealReal

Example Fourier Transform

Example Fourier Transform

Example Fourier Transform

Example Fourier TransformNote: FT is imaginary for sine

Example Fourier Transform Time Domain t Frequency Domain wRealRealDC component

FT of Delay d(t-t)Amplitude + phase is easier to understand:

(click movie)

Amplitude:Gives you information about frequencies/tones in a signal.Phase:More about when it happens in time.

Important FT PropertiesAddition

Scalar Multiplication

Convolution in time t

Convolution in frequency w

FT timefrequency duality

Time DomainFrequency DomainnarrowwidewidenarrowMultiplicationConvolution ConvolutionMultiplicationBoxSincSincBoxGaussGaussReal + EvenReal+Even (just cosine)Real + OddIm + Odd (just sine)Etc..Etc..

FT: Reframing the problem in Frequency DomainProblemx(t),h(t)Solutionof the equationResult*Fourier TransformInverseFourierTransformX(w), H(w) X(w)H(w)x= HARD= EASYCompletely sidesteps the convolution!

FT: Another ExampleWhat is the amplitude spectrum |Y(f)| of a voice signal (bandlimited to 5 kHz) when multiplied by a cosine f=15 kHz?(Note: this is Amplitude Modulation AM radio)

f

FT: Solution(Look Ma! No Algebra!)

t

x(t)=?

FT Gaussian Blur*=FrequencySpace

Sampling TheoremIn order to be used within a digital system, a continuous signal must be converted into a stream of values.

Done by sampling the continuous signal at regular intervals.

But at which interval?

Sampling TheoremSampling can be thought of multiplying a signal by a d pulse train:

t

x(t)

AliasingIf sampling rate is too small compared with frequency of signal, aliasing WILL occur:

Fourier Analysis of SamplingThe FT of a pulse train with frequency fs is another pulse train with interval 1/fs:

...

...

t

FT

...

...

f

fs

1/fs

Fourier Analysis of SamplingAliasing will happen if fs

A few sampling frequenciesTelephone systems: 8 kHz

CD music: 44.1 kHz

DVD-audio: 96 or 192 kHz

Aqua robot: 1 kHz

Digital Thermostat (HMTD84) : 0.2 Hz

Laplace TransformFormal definition:

Compare this to FT:

Small differences:Integral from 0 to to for Laplacef(t) for t

Common Laplace TransfomNamef(t)F(s)Impulse dStepRampExponentialSine1Damped Sine

Laplace Transform PropertiesSimilar to Fourier transform:Addition/Scaling

Convolution

Derivation

Transfer Function H(s)DefinitionH(s) = Y(s) / X(s)Relates the output of a linear system (or component) to its input.Describes how a linear system responds to an impulse.All linear operations allowedScaling, addition, multiplication.H(s)X(s)Y(s)

RC Circuit Revisited

*

t

=

t

x

step function

t

t

-

Time Domain

Laplace Domain

Poles and Zeros

Poles and Zeros

Poles and ZerosNamef(t)F(s)Impulse dStepRampExponentialSine1Damped SinePoles00 (double)n/a-a-iw,iw-a-iw,-a+iw

Poles and ZerosIf pole has:Real negative: exponential decayReal positive: exponential growthIf imaginary 0: oscillation of frequency w

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