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Unformatted text preview: EEL 5525: Foundations of DSP
Topic 2
DiscreteTime Signals and Systems
Dr. Liuqing Yang
Department of ECE
University of Florida
Email: lqyang@ece.ufl.edu
http://www.yang.ece.ufl.edu 1 Dr. Liuqing Yang Outline 2.1 Review of continuoustime signals
and systems
2.2 Discretetime signals
2.3 Discretetime systems 2
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Roadmap 2.1 Review of continuoustime signals
and systems
–
–
–
– Continuoustime signals
Fourier transform
Classification of continuoustime systems
Representation of LTI systems 2.2 Discretetime signals
2.3 Discretetime systems
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang ContinuousTime Signals • Dirac delta function • Sinc function • Gaussian function 4
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Fourier Transform
• Fourier transform (FT)
– Sufficient condition: • Inverse FT (IFT)
– Sufficient conditions (Dirichlet conditions):
• x(t) satisfies
• x(t) is continuous, except for discontinuity points whose
number on any finite interval is finite; the limits at both sides of each discontinuity point exist
• x(t) has a finite number of extrema on any finite interval
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Fourier Transform Pairs Delta vs. DC Sinc vs. Rect Gaussian function
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang The Impulse Train 7
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Systems
• Singleinput singleoutput (SISO) systems
• Memoryless systems • Linear SISO systems: additivity &
homogeneity
• Timeinvariant systems: shiftinvariant
• Linear timeinvariant (LTI) systems 8
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Representation of LTI Systems
• Impulse response
• Frequency response 9
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Roadmap 2.1 Review of continuoustime signals
and systems
2.2 Discretetime signals
– Sequences
– Operations of sequences
– Special sequences 2.3 Discretetime systems 10
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang DiscreteTime Signals: Sequences 11
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Example Sequences 12
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Operations on Sequences • Product:
• Sum:
• Scalar multiplication:
• Unit delay:
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang FIR Filter Example 14
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang IIR Filter Example 15
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang IIR Filter Example: Alternative Design 16
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang IIR Filter Simulation
Simulation for b=a=[1 1 1]
x[n] x[n1] x[n2] y[n] y[n1] y[n2] 1
2
3
4
5
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7 x = 1:10; b = [1 1 1]; a = [1 1 1]; y = filter(b,a,x)
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Upsample & Downsample • Upsample by a factor of L: ↑L – Insert (L1) zeros between every sample of
the input sequence.
– MATLAB: upsample(x,L) • Downsample by a factor of M: ↓M – Keep every Mth sample of the input
sequence.
– MATLAB: downsample(x,M)
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Special Sequences • Unit sample sequence • Delayed unit sample sequence 19
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Special Sequences • Unit step sequence • Delayed unit step sequence 20
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Special Sequences • Real sinusoidal sequence A: amplitude, w0: angular frequency, f: phase
• Period: Integer N ≥ 1 such that w0N=2pr,
where integer r ≥ 0.
• Fundamental Period: The smallest N which
satisfies the above condition.
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Random Sequence
• Generating random numbers in MATLAB
– rand(M,N) MxN matrix of uniformly
distributed random numbers between 0
and 1
– randn(M,N) MxN matrix of Gaussian
distributed random numbers with zero
mean and unit variance 22
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang >>
>>
>>
>> h = randn(8,1)+j*randn(8,1);
figure(1); plot(abs(h))
grid; axis([1 8 1 3])
xlabel('n'); ylabel('h[n]'); 3
2.5
2 h[n] 1.5
1
0.5
0
0.5
1 1 2 3 4 5 6 7 8 n 23
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang >>
>>
>>
>> x1 = sign(rand(256,1).5)
figure(2); plot(x1)
axis([0 256 1.5 1.5]);
xlabel('n'); ylabel('x_1[n]'); 1.5 1 1 x [n] 0.5 0 0.5 1 1.5 0 50 100 150
n 200 250 24
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Roadmap 2.1 Review of continuoustime signals
and systems
2.2 Discretetime signals
2.3 Discretetime systems
– Classification of discretetime systems
– Interconnected systems
– Linear constantcoefficient difference
equations 25
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Linear System • A system is linear if superposition holds:
– Suppose input x1[n] gives output y1[n] – and input x2[n] gives output y2[n].
– When the input is x[n]=ax1[n]+bx2[n]
– the system is said to be linear if the output
is y[n]=ay1[n]+by2[n]
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang TimeInvariant System
• Also known as (a.k.a.) ShiftInvariance
– Suppose input x[n] gives output y[n]
– When the input is x[nm]
– the system is said to be timeinvariant if
the output is y[nm] 27
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang LTI System
A system is said to be LTI if it is both linear and timeinvariant. • If input is the unit sample d[n], the
output is called the impulse response
h[n].
• If input is the unit step function m[n],
the output is called the step response
s[n].
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang LTI System Representation • An arbitrary input sequence • For an LTI system, superposition holds 29
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Examples of LTI Systems
• Ideal delay
• Moving average • Accumulator
• Forward difference • Backward difference
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Causal System
• A system is causal if the output sample
y[m] only depends on input samples
x[n] for n≤m. 31
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Stable System • Boundedinput boundedoutput (BIBO)
stable:
• For any
Q: When is an LTI system stable?
A: If and only if
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Cascade/Parallel System Connections • Cascade (a.k.a. Series) connection of
systems: • Parallel connection of systems 33
© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Linear Constant Coefficient
Difference Equations
• A class of LTI systems satisfying
• usually normalized such that d0=1 • Recursive solution • Generally IIR filter, unless dk=0, k • Total solution: y[n] = yc[n]+yp[n] • yc[n]: complementary/homogeneous solution
• yp[n]: particular/forcing solution
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Auxiliary Conditions in Linear
ConstantCoefficient Difference Equations
• Required to uniquely determine the output for
a given input.
• Can be given as N sequential values of the
output.
• Can be used to recursively compute later and
prior values of the output.
• Determines the L, TI and causality of the
system.
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© Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Summary
• FT of an Impulse Train is an Impulse Train
• Some special discretetime signals
– Fundamental period of discretetime sinusoidal signals • Discretetime systems:
– I/O relationship,
– Linearity, Time invariance
– Causality, Stability • Linear constant coefficient difference equations
– Recursive solution
– Total solution
– Auxiliary conditions • Some MATLAB functions
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 Fall '09
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