Lecture 16 Outline:
Discrete Fourier Series and Transforms
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Announcements:
Reading: “5: The Discrete Fourier Transform” pp. 1-9.
 HW 5 posted, short HW (2 analytical and 1 Matlab problem), due Wed 5pm.
No late HWs as solutions will be available immediately.
 Midterm details on next page
 HW 6 will be posted Wed, due following Wed with free extension to Thurs.
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Review of Last Wed. Lecture
Discrete Fourier Series
Discrete Fourier Transform
Relation between DFT & DTFT
DFT as a Matrix Operation
Properties of DFS and DFT
Midterm Details

Time/Location: this Friday, May 6, 9:20am-11:20am in this room.
For students with 9:30am classes, can take 10:30-12:30 in 200-203 (two floors up)
 We should be aware of all conflicts with the 9:20-11:20am timeslot at this time
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Open book and notes – you can bring any written material you wish to
the exam. Calculators and electronic devices not allowed.
Will cover all class material from Lectures 1-14.
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See lecture ppt slides for material in the reader that you are responsible for
Practice MT will be posted today, worth 25 extra credit points for
“taking” it (not graded).
Can be turned in any time up until you take the exam
 Solutions given when you turn in your answers
 In addition to practice MT, we will also provide additional practice problems/solns
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Instead of MT review, we will provide extra OHs for me and the TAs
to go over course material, practice MT, and practice problems
My extra OHs: Wed 2-3:30pm, Thurs 3-4:30pm.
 TA extra OHs will be announced and posted on calendar later today (mostly WTh)
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Review of Last Wed Lecture
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FIR design entails choice of window function to mitigate Gibbs

Goal is to approximated desired filter without Gibbs/wiggles
Design tradeoffs involve main lobe vs. sidelobe sizes
 Typical windows: rectangle (boxcar), triangle, Hanning, and Hamming
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FIR design for desired hd[n] entails picking a length M, setting
ha[n]=hd[n], |n|M/2, choosing window w[n] with hw[n]=h[n]w[n]to
mitigate Gibbs, and setting h[n]=hw[n-M/2] to make design causal
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FIR implemented directly using M delay elements and M+1 multipliers
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Can introduce group delay
Efficiently implemented with DFT
Example design for LFP (Differentiator in HW)
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Hamming smooths out wiggles from rectangular window
Introduces more distortion at transition frequencies than rectangular window
Discrete Fourier Series

The DFS is the DTFS with a different normalization:

Consider an N-periodic discrete-time signal ~x n:
~
x n  N   ~
x n  n


Then
.
•Usi
~
X k   N  a k
n
Define WN ≜ e

j
ak 
1
N
 ~x [n]e
 jk ( 2 / N ) n
n  N 
is also N-periodic: X~ k  N   X~ k  k
2
N
Appears ing the DTFS, DFS or DFT for N-periodic sequences
this
nota
tion,
we
writ
e
j
2
nk
N

Then we can write WNkn  e

Using this notation, we have the DFS pair for periodic signals:
1
~
x n  
N

N 1

k 0
~
X k WNkn
~
X k  
N 1
 ~x nW
kn
N
n 0
Simple computation of WN kn makes this pair easier to compute than DTFS
Discrete Fourier Transform (DFT)

Works with only one period of ~x n and X~ k 
x n 0  n  N  1
~
xn  
otherwise
 0

Can recover original periodic sequences ~x n, X~ k  as

~
x n  xn  N  

xn  rN 
r  


~
 X k  0  k  N  1
X k   
otherwise
 0
~
X k   X k  N  

n N
 n mod N
r  
k N
 k mod N
 X k  rN 
Equivalently, work with N samples of x[n]
Conjugate Relationships
Leads to DFT Pair
1
xn  
N
N 1
 X k 
WNkn
k 0
Inverse DFT
X k  

N 1
 xnW
n 0
DFT
 

DFT X * k   N DFT 1 X k 
kn
N
DFT 1 X k  


*

1
DFT X * k 
N
DFT/IDFT commonly used in DSP, using N-length signal blocks, due to its
much lower computational complexity than the DTFT/IDTFT
*
Example
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Real and even
~
~
x n and X k 
~
X k 
~
x n 

N 1
n
N 1
0
...
...
...
...
k
0
One-period representations:
X k 
xn 
N 1
0
N 1
n
k
0
•or
Relation between DFT and DTFT
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Given a length-N signal x[n], n-point DFT is
N 1
X k    xn W
n 0

kn
N
N 1
  xn e
j
2
nk
N
, 0  k  N 1
n 0
Its DTFT is
X e
j
N 1
   xne
 jn

n 0

DFT is the DTFT sampled at N equally spaced
frequencies between 0 and 2:
X k   X e j 
k
2
N
, 0  k  N 1
DFT/IDFT as Matrix Operation
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
DFT
 X 0   W N00

 


  
 X k     W k 0
N

 


  

   N 10
 X N  1 W N

W N0 n





W Nkn




 WN

Inverse DFT
 W N00
 x0 





  
 xn    1  W  n 0
N

 N

  

   N 10


 xN  1
W N

 N 1n
W N0 N 1  x0 







k  N 1 
WN
 xn 




 N 1 N 1  xN  1
WN



W N0 k





W N nk





W N  N 1k

W N0 N 1  X 0 







 n  N 1 
WN
 X k  





  N 1 N 1 
 X N  1
WN


Computational Complexity
of an N-point DFT or inverse
DFT requires N 2 complex multiplications.
 Computation
Properties of the DFS/DFT
Properties (Continued)
Main Points
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DFS is the DTFS with a different normalization
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DFT operates on one N-length “piece” of a signal x[n]
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Fast/low complexity computation (N2 complex multiplications)

DFT is the DTFT sampled at N equally spaced
frequencies between 0 and 2

DFT/IDFT can be calculated via a matrix multiplication

DFS/DFT have similar properties as DTFS/DTFT but
with modifications due to periodic/circular characteristics