elec361: signals and systems topic 3: fourier series...

o Introduction to frequency analysis of signalso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to complex exponentialo Summary o Appendix:

oApplications (not in the exam)

ELEC361: Signals And Systems

Topic 3: Fourier Series (FS)

Dr. Aishy AmerConcordia UniversityElectrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

•M.J. Roberts, Signals and Systems, McGraw Hill, 2004

•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

Signal Representation

Time-domain representationWaveform basedPeriodic / non-periodic signals

Frequency-domain representationPeriodic signalsSinusoidal signalsFrequency analysis for periodic signalsConcepts of frequency, bandwidth, filtering

Waveform Representation

Waveform representationPlot of the signal value vs. time

Sound amplitude, temperature reading, stock price, ..

Mathematical representation: x(t)x: variable valueT: independent variable

Sample Speech Waveform

Sample Music Waveform

Sinusoidal Signals

Sinusoidal signals: important because they can be used to synthesize any signalAn arbitrary signal can be expressed as a sum of many sinusoidal signals with different frequencies, amplitudes and phasesPhase shift: how much the max. of the sinusoidal signal is shifted away from t=0

Music notes are essentially sinusoids at different frequencies

Complex Exponential Signals

)sin(2)cos(2:ForumlaEuler

)()(:Note:ConjugateComplex

phase) (initialshift phase theis )sin()cos(||)(

:Signal lExponentiaComplex sincos :tionrepresentaPolar

tan is z of Phase

|| is z of Magnitude

:tionrepresentaCartesian :NumberComplex

θθωθω

θθθθ

realarezzandzzjbaz

tzjtzeztx

zjzezzabz

=−=+

−−

Real and Complex Sinusoids

Periodic CT Signals

A CT signal is periodic if there is a positive value for which

Period T of : The interval on which x(t) repeatsFundamental period T0: the smallest such repetition interval T0 =1/f0Fundamental period: the smallest positive value for which the equation above holdsExample: x(t) = cos(4*pi*t); T=1/2; T0=1/4Harmonic frequencies of x(t): kf0 , k is integer,

Example: is periodic with

Sums of CT periodic signals

The period of the sum of CT periodic functions is the least common multiple of the periods of the individual functions summed

If the least common multiple is infinite, the sum is aperiodic

Periodic DT Signals

A DT signal is periodic with period where is a positive integer if

The fundamental period of is the smallest positive value of for which the equation holdsExample:

is periodic with fundamental period

What is frequency of an arbitrary signal?

Sinusoidal signals have a distinct (unique) frequencyAn arbitrary signal x(t) does not have a unique frequencyx(t) can be decomposed into many sinusoidal signals with different frequencies, each with different magnitude and phaseSpectrum of x(t): the plot of the magnitudes and phases of different frequency componentsFourier analysis: find spectrum for signalsBandwidth of x(t): the spread of the frequency components with significant energy existing in a signal

Frequency content in signals

Frequency content in signalsA constant : only zero frequency component (DC component)A sinusoid : Contain only a single frequency componentPeriodic signals : Contain the fundamental frequency and harmonics : Line spectrumSlowly varying : contain low frequency onlyFast varying : contain very high frequencySharp transition : contain from low to high frequencyMusic: :

contain both slowly varying and fast varying components, wide bandwidth

Transforming Signals

X(t) is the signal representation in time domainOften we transform signals in a different domain Fourier analysis allows us to view signals in the frequency domainIn the frequency domain we examine which frequencies are present in the signalFrequency domain techniques reveal things about the signal that are difficult to see otherwise in the time domain

Fourier representation of signals

The study of signals and systems using sinusoidal representations is termed Fourier analysis, after Joseph Fourier (1768-1830) The development of Fourier analysis has a long history involving a great many individuals and the investigation of many different physical phenomena, such as the motion of a vibrating string, the phenomenon of heat propagation and diffusionFourier methods have widespread application beyond signals and systems, being used in every branch of engineering and scienceThe theory of integration, point-set topology, and eigenfunction expansions are just a few examples of topics in mathematics that have their roots in the analysis of Fourier series and integrals

Fourier representation of signals: Types of signals

Fourier representation of signals: Continuous-Value / Continuous-Time Signals

All continuous signals are CT but not all CT signals are continuous

Fourier representation of signals: Types of signals

Fourier representation of signals

Four distinct Fourier representations:Each applicable to a different class of

signalsDetermined by the periodicity properties of the signal and whether the signal is discrete or continuous in time

A Fourier representation is unique, i.e., no two same signals in time domain give the same function in frequency domain

Overview of Fourier Analysis Methods

Discrete in Time

Periodic in Frequency

Continuous in Time

Aperiodic in Frequency

Aperiodic in TimeContinuous in Frequency

Periodic in TimeDiscrete in Frequency

−∞=

⇒⊗

dtetxT

P-CTDT :SeriesFourier Inverse CT

DTP-CT :SeriesFourier CT

⇒+⊗

+⇒⊗∞

−∞=

ωπ 2

)(21][

DT PCT :TransformFourier DT Inverse

PCTDT :TransformFourier DT

deeXnx

⇒⊗

P-DTP-DT SeriesFourier DT Inverse

P-DTP-DT SeriesFourier DT

∞−

⇒⊗

ωωπ

dejXtx

dtetxjX

)(21)(

CTCT :TransformFourier CT Inverse

CTCT :TransformFourier CT

Fourier representation: Periodic Signals

2zzR(z)

spectrum) (symmetric :signals realFor

numbercomplex a general,in is,

,...2,1,0;)(1][

) transform(forward Analysis SeriesFourier

complex) and realboth for sided, (double

only) signal realfor sided, (single )cos()(

transform)(inverser Synthesis SeriesFourier

−−

−∞=

±±===

kdtetxT

ktaatx

FS: Example

Componentsconstant -NonComponent Constant )(.4.4

.7.710)(

500cos(8)3

200cos(1410)(

2502225022

1002310023

+=⇒++

++−+=

−−

txeeee

eeeetx

tjjtjj

ππππ

Concept of Fourieranalysis

Concept of Fourier analysis

Approximation of Periodic Signals by Sinusoids

Any periodic signal can be approximated by a sum of many sinusoids at harmonic frequencies of the signal (kf0 ) with appropriate amplitude and phaseThe more harmonic components are added, the more accurate the approximation becomesInstead of using sinusoidal signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies

Approximation of Periodic Signals by Sum of Sinusoids

Fourier representation: Periodic CT Signals

Example: Fourier Series of Square Wave

The Fourier series analysis:

Example: Spectrum of Square Wave

Each line corresponds to one harmonic frequency. The line magnitude (height) indicates the contribution of that frequency to the signalThe line magnitude drops exponentially, which is not very fast. The very sharp transition in square waves calls for very high frequency sinusoids to synthesize1

Negative Frequency?

Why Frequency Domain Representation of signals?

Shows the frequency composition of the signalChange the magnitude of any frequency component arbitrarily by a filtering operation

Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionHigh emphasis -> edge enhancement

Shift the central frequency by modulationA core technique for communication, which uses modulation to multiplex many signals into a single composite signal, to be carried over the same physical medium

Why Frequency Domain Representation of signals?

Typical Filtering applied to x(t):Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionBandpass -> Retain only a certain frequency range

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to complex exponentialo Summary

Fourier series of CT periodic signals

Consider the following continuous-time complex exponentials:

T0 is the period of all of these exponentials and it can be easily verified that the fundamental period is equal to

Any linear combination of is also periodic with period T0

Fourier series of CT periodic signals

fundamental components or the first harmonic components

The corresponding fundamental frequency is ω0

Fourier series representation of a periodic signal x(t):

Fourier series of a CT periodic signal

Fourier series of a CT periodic signal: Example 4.1

The Fourier series coefficients are shown in the Figures Note that the Fourier coefficients are complex numbers in generalThus one should use two figures to demonstrate them completely: show

real and imaginary parts ormagnitude and angle

If x(t) is real

This means that

Fourier series of CT REAL periodic signals

“sinc” Function

Fourier series of a CT periodic signal:Example 4.2

Fourier series of a CT periodic signal: Example 4.2

The Fourier series coefficients are shown in Figures for T=4T1 and T=16T1Note that the Fourier series coefficients for this particular example are real

Inverse CT Fourier Series: Example: Magnitude and Phase Spectra of the harmonic function X[k]

Inverse CT Fourier series: Example

The CT Fourier Series representation of the above cosinesignal X[k] is

is odd The discontinuities make X[k] have significant higher harmonic content

Note: Log-Magnitude Frequency Response Plots

Outline

Effect of Signal Symmetry on CT Fourier Series

Effect of Signal Symmetry on CT Fourier Series: Example

Outline

Properties of CT Fourier series

The properties are useful in determining the Fourier series or inverse Fourier seriesThey help to represent a given signal in term of operations (e.g., convolution, differentiation, shift) on another signal for which the Fourier series is knownOperations on {x(t)} Operations on {X[k]}Help find analytical solutions to Fourier Series problems of complex signals Example:

tionmultiplicaanddelaytuatyFS t →−= })5()({

Let x(t): have a fundamental period T0x

Let y(t): have a fundamental period T0y

Let X[k]=ak and Y[k]=bkThe Fourier Series harmonic functions each using the fundamental period TF as the representation time

In the Fourier series properties which follow:Assume the two fundamental periods are the same T= T0x =T0y (unless otherwise stated)

The following properties can easily been shown using equation (4.5) for Fourier series

Properties of CT Fourier series: Example 5.1

Properties of CT Fourier series: Example 5.2

Properties of CT Fourier series: Example

Outline

Convergence of the CT Fourier series

The Fourier series representation of a periodic signal x(t) converges to x(t) if the Dirichlet conditions are satisfiedThree Dirichlet conditions are as follows:

1. Over any period, x(t) must be absolutely integrable.

For example, the following signal does not satisfy this condition

2. x(t) must have a finite number of maxima and minima in one periodFor example, the following signal meets Condition 1, but not

Condition 2

3. x(t) must have a finite number of discontinuities, all of finite size, in one periodFor example, the following signal violates Condition 3

Every continuous periodic signal has an FS representationMany not continuous signals has an FS representationIf a signal x(t) satisfies the Dirichlet conditions and is not continuous, then the Fourier series converges to the midpoint of the left and right limits of x(t) at each discontinuityAlmost all physical periodic signals encountered in engineering practice, including all of the signals with which we will be concerned, satisfy the Dirichletconditions

Convergence of the CT Fourier series: Summary

Convergence of the CT Fourier series: Continuous signals

Convergence of the CT Fourier series: Discontinuous signals

Convergence of the CT Fourier series: Gibb’s phenomenon

Convergence of the CT Fourier series: Gibbs Phenomenon

Outline

The DT Fourier Series

Concept of DT Fourier Series

The DT Fourier Series

The DT Fourier Series: Example 5.3

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier serieso Response of LTI systems to complex exponentialo Summary

Properties of DT Fourier Series

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to Complex Exponentialo Summary

Response of LTI systems to Complex Exponential

Eigen-function of a linear operator S:a non-zero function that returns from the operator exactly as is except for a multiplicator(or a scaling factor)

Eigenfunction of a system S: characteristic function of S

function-eigen :)( vector)null-non (a value-eigen :

)()}({:)(function aon applied System

txtxStxλ

Eigenfunction of an LTI system = the complex exponentialAny LTI system S excited by a complex sinusoid responds with another complex sinusoid of the samefrequency, but generally a different amplitude and phaseThe eigen-values are either real or, if complex, occur in complex conjugate pairs

tj ke ω

Convolution represents:The input as a linear combination of impulses

The response as a linear combination of impulse responses

Fourier Series represents:a periodic signal as a linear combination of complex

sinusoids

∫∞

∞−

−= ττδτ dtxtx )()()(

∫∞

∞−

−= τττ dthxty )()()(

022)( fkfeatx kkk

k ππωω === ∑∞

−∞=

Response of LTI systems to Complex Exponential : Linearity and Superposition

If x(t) can be expressed as a sum of complex sinusoids the response can be expressed as the sum of responses to complex sinusoids

tjk febty k πωω 2)( == ∑

−∞=

Let a continuous-time LTI system be excited by a complex exponential of the form,

The response is the convolution of the excitation with the impulse response or

The quantity

will later be designated the Laplace transform of the impulse response and will be an important transform method for CT systemanalysis

CT system:

This leads to the following equation for CT LTI systems:

ωσωσ jseetx tjst +=== + ;)( )(

H(s) is a complex constant whose value depends on s and is given by:

Complex exponential est are eigenfunctions of CT LTI systemsH(s) is the eigenvalue associated with the eigenfunction est

DT systems:

This leads to the following equation for DT LTI systems:

H(z) is a complex constant whose value depends on z and is given by:

Complex exponential zn are eigenfunctions of DT LTI systemsH(z) is the eigenvalue associated with the eigenfunction zn

From superposition:

Response of LTI systems to Complex Exponential: Example 3.3

Consider an LTI system whose input x(t) and output y(t) are related by a time shift as follows:

Find the output of the system to the following inputs:

Example 3.3 - Solution

Response of LTI systems to Complex Exponential : ωω jsetx tj +== 0;)(

Response of LTI systems to Complex Exponential: Example

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to periodic signalso Summary

Fourier series: summary

Sinusoid signals:Can determine the period, frequency, magnitude and phase of a sinusoid signal from a given formula or plot

Fourier series for periodic signalsUnderstand the meaning of Fourier series representationCan calculate the Fourier series coefficients for simple signals (only require double sided)Can sketch the line spectrum from the Fourier series coefficients

Fourier series: summary

Steps for computing Fourier series:1. Identify period2. Write down equation for x(t)3. Observe if the signal has any

summitry (even or odd)4. Use the exponential equation (1) in

previous slide, and if needed use Eq. (2) for the trigonometric coefficients

FS Summary: a quizProblem: Find the Fourier series coefficients of the periodic continuous signal: and is the period of the signal. Plot the spectrum (magnitude and phase) of x(t). What is the spectrum of x(t-4)?

Solution: Using the definition of Fourier coefficients for a periodic continuous signal, the coefficients are:

which would be simplified after some manipulations to: …

cos()( <≤= tttx π3=T

∫∫∫−

−−− +

0 231)

0 dteeedtetdtetxT

tjtjtjkT

πππ

πω π

2kkjak −

=⇒π

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to complex exponential o Summaryo Appendix: Applications (not in the exam)

Applications of frequency- domain representation

Clearly shows the frequency composition a signalCan change the magnitude of any frequency component arbitrarily by a filtering operation

Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionHigh emphasis -> edge enhancement

Can shift the central frequency by modulationA core technique for communication, which uses modulation to multiplex many signals into a singlecomposite signal, to be carried over the same physical medium

Processing of speech and music signals

Typical Filters

Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionBandpass -> Retain only a certain frequency range

Low Pass Filtering(Remove high freq, make signal smoother)

High Pass Filtering(remove low freq, detect edges)

Filtering in Temporal Domain(Convolution)

Communication: Signal Bandwidth

Bandwidth of a signal is a critical feature when dealing with the transmission of this signalA communication channel usually operates only at certain frequency range (called channel bandwidth)

The signal will be severely attenuated if it contains frequencies outside the range of the channel bandwidthTo carry a signal in a channel, the signal needed to be modulated from its baseband to the channel bandwidthMultiple narrowband signals may be multiplexed to use a single wideband channel

Signal bandwidth

Highest frequency estimation in a signal:Find the shortest interval between

peak and valleys

Signal Bandwidth

Estimation of Maximum Frequency

Processing Speech & Music Signals

Typical speech and music waveforms are semi-periodicThe fundamental period is called pitch periodThe fundamental frequency (f0)

Spectral contentWithin each short segment, a speech or music signal can be decomposed into a pure sinusoidal component with frequency f0, and additional harmonic components with frequencies that are multiples of f0.The maximum frequency is usually several multiples of the fundamental frequencySpeech has a frequency span up to 4 KHzAudio has a much wider spectrum, up to 22KHz

Sample Speech Waveform 1

Numerical Calculation of CT Fourier Series

The original signal is digitized, and then a Fast Fourier Transform (FFT) algorithm is applied, which yields samples of the FT at equally spaced intervalsFor a signal that is very long, e.g. a speech signal or a music piece, spectrogram is used.

Fourier transforms over successive overlapping short intervals

Sample Speech Spectrogram 1

Sample Speech Waveform 2

Speech Spectrogram 2

Sample Music Waveform

Sample Music Spectrogram

elec361: signals and systems topic 3: fourier series...

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