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o Topic 1: LTI Signals & Systemso Topic 2: Convolutiono Topic 3: Fourier Serieso Topic 4: Fourier Transform
ELEC361: Signals And Systems
Midterm Review
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
Topic 1:LTI Signals & Systems
Topic 1:LTI Signals & Systems
Topic 1:LTI Signals & Systems
Topic 1:LTI Signals & Systems
Topic 2: Convolution
Topic 2: Convolution
Types of SystemsTime-invariant, linear, causal, memory, stable, invertibleNeed to be able to prove whether a system is linear and time-invariant
Convolution Sum Know how to compute impulse response h[n]
Convolution Integral Graphical Convolution
Convolution PropertiesStability and Impulse Response
Topic 2: Convolution
Convolution:
1. Write down equations for both signals2. Reflect and shift one of the two signals (preferably the finite
duration signal)3. Label the edge(s) of the reflected/shifted signal with
t and/or t-a appropriately 4. Shift signal to the left until there is no overlap between the two
signals5. Identify number of different cases in which there is overlap
between the two signals6. For each of the above cases, evaluate the convolution integral
with the limits of integration determined by the region of overlap expressed as a function of t ( see step 3)
Topic 2: Convolution
Topic 2: Convolution
Topic 2: Convolution
Topic 3: Fourier Series
Topic 3: Fourier SeriesFourier Series:
Exponential Fourier series
Trigonometric Fourier Series∑
∫∞
−∞=
−
=
==
k
tkfj
T
tkfj
ekXtx
TfdtetxT
kX
0
0
2
02
)()(
)1(/1)(1)(
π
π
∑∑∞
=
∞
=
++=
==−===
10
100
0
)2sin()2cos()(
)0()( that note)}(Im{2
)2()( that note)}(Re{2
kk
kk
ksk
kck
tkfbtkfaatx
XabkXkXb
akXkXa
ππ
Topic 3: Fourier Series
Fourier SeriesSymmetries of signal (even and odd)
• Even signal: X(k) is purely real• Odd signal: X(k) is purely imaginary
Properties of Fourier seriesShifting property
)()( 0020 kXettx tkfj π−↔−
Topic 3: Fourier Series
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
Topic 3: Fourier Series
Topic 3:Fourier Series
Topic 4: Fourier Transform
Topic 4: Fourier Transform
Fourier transform for aperiodic signalsUnderstand the meaning of the inverse Fourier transformSketch the spectrumDetermine the bandwidth of the signal from its spectrumKnow how to interpret a spectrogram plot
Topic 4: Fourier Transform
Topic 4: Fourier Transform
Topic 4: Fourier Transform
Topic 4: Fourier Transform
Topic 4: Fourier Transform
