Lecture 5: Test 1 review
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Transcript Lecture 5: Test 1 review
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Lecture 5: Test 1 review
Instructor:
Dr. Gleb V. Tcheslavski
Contact: [email protected]
Office Hours: Room 2030
Class web site:
http://ee.lamar.edu/gleb/dsp/ind
ex.htm
ELEN 5346/4304
DSP and Filter Design
Fall 2008
2
Fundamental system’s properties
(5.2.1)
1. Linearity:
x1n y1n andx2n y2n ax1n bx2n ay1n by2n x1n ,x2n ,a,b
2. Time-invariance:
xn yn xnn0 ynn0 xn n0
(5.2.2)
3. Causality:
xn yn yn fct ( xk k n) -nooutputbeforeinput
(5.2.3)
4. Stability (BIBO):
xn Bx yn By for every bounded input,
(5.2.4)
the output must be bounded
1,2,3 Must hold for ANY inputs and constants!
Most of properties are specified for the relaxed systems (empty states)!
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Fall 2008
3
Difference equations
LCCDELTI system
states
xn
yn
memory
No driving terms – homogeneous solution.
LTI
Whatever is stored (represents the memory of the system) operations on the
previous inputs initial conditions reflected in the homogeneous solution.
Usually:
ytot ,n yh,n y p,n ;n n0
(5.3.1)
Remember and understand SFGs of an LTI system.
ytot ,n yzi,n yzs,n
zero-input
(5.3.2)
zero-state
ytot ,n ytr ,n yss,n
transient
(5.3.3)
steady-state
The connection between (5.2.1), (5.2.2), and (5.2.3) is the same (in general) total response
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Fall 2008
4
Difference equations
Example: find the total solution for the system specified by the following DE:
yn yn1 6 yn2 xn
For the following input and the initial conditions:
xn 8un
y1 1; y2 1
Let’s find the homogeneous solution first: xn = 0. The characteristic eqn.:
n n1 6 n2 0 n2 ( 2 6) n2 ( 3)( 2) 0
Characteristic roots are 1 = -3; 2 = 2. Therefore, the homogeneous solution is:
yh ,n C1 3 C2 2
n
For the particular solution, we assume:
Therefore, combining with DE for large n:
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n
y p,n k
k k 6k 8un
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Difference equations
Which gives k = -2.
Then, the total solution is:
yn C1 3 C2 2 2;n 0
n
n
We find the constants by satisfying the initial conditions:
2
2
y
C
3
C
2
2 1
1
2
2
C1 1.8;C2 4.8
1
1
y1 C1 3 C2 2 2 1
Finally, the total solution is:
yn 1.8 3 4.8 2 2;n 0
n
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n
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Difference equations
Alternatively, the zero-input response can be found as a form of the homogeneous
solution, where the constants are chosen to satisfy the initial conditions.
y0 y1 6 y2 1 6 7
y1 y0 6 y1 7 6 13
From the equation for the homogeneous solution, we get:
y0 C1 C2 ; y1 3C1 2C2
Therefore: C1 = -5.4; C2 = -1.6
The zero-input response is
yzi ,n 5.4 3 1.6 2 n 0
n
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Fall 2008
n
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Difference equations
The zero-state response can be found as a form of the total solution when zero
initial conditions are satisfied.
y0 x0 8; y1 x1 y0 0
We find the constants for the total solution as C1 = 3.6 and C2 = 6.4. Therefore,
the zero-state response for n 0 and with i.c.s yzs,-2 = yzs,-1 = 0 is:
yzs ,n 3.6 3 6.4 2
n
n
Again, the total solution is:
yn 1.8 3 4.8 2 2;n 0
n
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n
Fall 2008
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LTI systems
If thesystem ' sinput :xn xk nk
(5.4.1)
k
If the system is linear:
thesystem ' soutput : yn xk ( nk ) {and TI } xk hnk
k
since
(5.4.2)
k
n hn ;nk hn,k (ingeneral ).ForTI :nk hnk
(5.4.3)
Therefore:
yn xk hnk hk xnk xn hn hn xn linearconvolution
k
(5.4.4)
k
hn yzs,n whenxn n
Alternatively:
(5.4.5)
LTI ,BIBO hk Bh
(5.4.6)
LTI ,Causalhn hnun (hn 0n 0)
(5.4.7)
k
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SFG
Let a yn k bm xn-m
(5.5.1)
m
S n 1 AS n bxn
In general!
(5.5.2)
( A) 1 BIBO
(5.5.3)
yn cT S n dxn
(5.5.4)
if xn 0andnislargeSn 0 yn 0
- Relaxed system
If there is no input, the output must go to zero!
What’s about unstable AND not-relaxed systems?
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(5.5.5)
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SFG
Signal flow graph does not necessarily have to represent
Fundamental of Direct 2 forms! They can be arbitrary…
z-1
d1
A perfectly valid
form of SFG:
xn
c1
c2
d2
d3
Find the LCCDE and statespace representation
quantities!
Regrouping an SFG might be a good idea!
Unit step response: empty states, impulse at the input.
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yn
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DTFT and DFT
DTFT of a finite length sequence (sampled at k
DTFT at
k
2 k
N
2
k ) is DFT{xn}
N
periodicxn xn rN
IDTFT
r
k 0,...N frequencysamples
generally, aliased result
“sampling” usually leads to “aliasing”!
IDFT{XkHk} =
xn hn
- a circular convolution
IDFT{XkHk} =
xn hn
- a linear convolution – a practical application
N
after sufficient zero-padding
“FFT” really means “Fast DFT”.
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Suggestions
You may need proofs, derivations, properties…
Look at the examples covered in class.
Think about “good questions” for the test!
SFG {A,b,c,d}
Need to understand and be able to test for:
Linearity, Time Invariance, BIBO, Causality.
properties?
What are and how to find:
yss,n, ytr,n, yzi,n, yzs,n, yh,n, yp,n?
Direct 1 and Direct 2 – implementation aspects
Be able to interpret results of DFT,
What are the frequency response (LTI), Impulse response, transient response?
Purpose of scaling??
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DSP and Filter Design
Fall 2008