ELEC361: Signals And Systems Topic 3: Fourier Series...
Transcript of 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
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Signal Representation
Time-domain representationWaveform basedPeriodic / non-periodic signals
Frequency-domain representationPeriodic signalsSinusoidal signalsFrequency analysis for periodic signalsConcepts of frequency, bandwidth, filtering
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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
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Sample Speech Waveform
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Sample Music Waveform
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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
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Complex Exponential Signals
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Real and Complex Sinusoids
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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
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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
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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
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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
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Frequency content in signals
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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
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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
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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
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Fourier representation of signals: Types of signals
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Fourier representation of signals: Continuous-Value / Continuous-Time Signals
All continuous signals are CT but not all CT signals are continuous
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Fourier representation of signals: Types of signals
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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
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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
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Fourier representation: Periodic Signals
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FS: Example
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Concept of Fourieranalysis
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Concept of Fourier analysis
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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
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Approximation of Periodic Signals by Sum of Sinusoids
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Fourier representation: Periodic CT Signals
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Example: Fourier Series of Square Wave
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The Fourier series analysis:
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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
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Negative Frequency?
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Negative Frequency?
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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
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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
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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
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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
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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):
(4.1)
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Fourier series of a CT periodic signal
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Fourier series of a CT periodic signal
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Fourier series of a CT periodic signal: Example 4.1
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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
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If x(t) is real
This means that
Fourier series of CT REAL periodic signals
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Fourier series of CT REAL periodic signals
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“sinc” Function
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Fourier series of a CT periodic signal:Example 4.2
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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
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Inverse CT Fourier Series: Example: Magnitude and Phase Spectra of the harmonic function X[k]
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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
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Note: Log-Magnitude Frequency Response Plots
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Note: Log-Magnitude Frequency Response Plots
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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
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Effect of Signal Symmetry on CT Fourier Series
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Effect of Signal Symmetry on CT Fourier Series
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Effect of Signal Symmetry on CT Fourier Series
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Effect of Signal Symmetry on CT Fourier Series
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Effect of Signal Symmetry on CT Fourier Series
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Effect of Signal Symmetry on CT Fourier Series: Example
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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
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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:
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Properties of CT Fourier series
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
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series
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Properties of CT Fourier series: Example 5.1
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Properties of CT Fourier series: Example 5.2
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Properties of CT Fourier series: Example 5.2
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Properties of CT Fourier series: Example
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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
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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
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Convergence of the CT Fourier series
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
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Convergence of the CT Fourier series
3. x(t) must have a finite number of discontinuities, all of finite size, in one periodFor example, the following signal violates Condition 3
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Convergence of the CT Fourier series
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
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Convergence of the CT Fourier series: Summary
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Convergence of the CT Fourier series: Continuous signals
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Convergence of the CT Fourier series: Discontinuous signals
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Convergence of the CT Fourier series: Gibb’s phenomenon
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Convergence of the CT Fourier series: Gibbs Phenomenon
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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
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The DT Fourier Series
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The DT Fourier Series
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The DT Fourier Series
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Concept of DT Fourier Series
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The DT Fourier Series
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The DT Fourier Series
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The DT Fourier Series
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The DT Fourier Series: Example 5.3
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The DT Fourier Series: Example 5.3
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The DT Fourier Series: Example 5.3
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The DT Fourier Series: Example 5.3
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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
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Properties of DT Fourier Series
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Properties of DT Fourier Series
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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
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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)
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Response of LTI systems to Complex Exponential
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
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Response of LTI systems to Complex Exponential
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
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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
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Response of LTI systems to Complex Exponential
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
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Response of LTI systems to Complex Exponential
CT system:
This leads to the following equation for CT LTI systems:
ωσωσ jseetx tjst +=== + ;)( )(
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Response of LTI systems to Complex Exponential
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
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Response of LTI systems to Complex Exponential
DT systems:
This leads to the following equation for DT LTI systems:
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Response of LTI systems to Complex Exponential
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
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Response of LTI systems to Complex Exponential
From superposition:
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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:
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Example 3.3 - Solution
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Example 3.3 - Solution
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Response of LTI systems to Complex Exponential : ωω jsetx tj +== 0;)(
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Response of LTI systems to Complex Exponential: Example
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Response of LTI systems to Complex Exponential: Example
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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
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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
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Fourier series: summary
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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
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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: …
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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)
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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
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Typical Filters
Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionBandpass -> Retain only a certain frequency range
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Low Pass Filtering(Remove high freq, make signal smoother)
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High Pass Filtering(remove low freq, detect edges)
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Filtering in Temporal Domain(Convolution)
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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
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Signal bandwidth
Highest frequency estimation in a signal:Find the shortest interval between
peak and valleys
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Signal Bandwidth
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Estimation of Maximum Frequency
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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
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Sample Speech Waveform 1
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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
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Sample Speech Spectrogram 1
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Sample Speech Waveform 2
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Speech Spectrogram 2
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Sample Music Waveform
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Sample Music Spectrogram