Lecture 15 Outline: DFT Properties and Circular Convolutionl Announcements:

l HW 4 posted, due Tues May 8 at 4:30pm. No late HWs as solutions will be available immediately.

l Midterm details on next pagel HW 5 will be posted Fri May 11, due following Fri (as usual)

l Review of Last Lecturel DFT as a Matrix Operationl Properties of DFS and DFTl Circular Time/Freq. Shift and Convolutionl Circular Convolution Methodsl Linear vs. Circular Convolution

Midterm Detailsl Time/Location: Friday, May 11, 1:30-2:50pm in this room.

l Open book and notes – you can bring any written material you wish to the exam. Calculators and electronic devices not allowed.

l Will cover all class material from Lectures 1-13.

l Practice MT posted, worth 25 extra credit points for “taking” it.l Can be turned in any time up until you take the exam (send scanned version to TAs, or

give them a hard copy in OHs/section)l Solutions given when you turn in your answers l In addition to practice MT, we will also provide additional practice problems/solns

l MT Review in class May 7

l Discussion Section May 8, 4:30-6 (MT review and practice problems)

l Regular OHs for me/TAs this week and next (no new HW next week)l I am also available by appointment

Review of Last Lecturel Discrete Fourier Series (DFS) Pair for Periodic Signals

l Discrete Fourier Transform (DFT) Pair

l and are one period of and , respectively

l DFT is DTFT sampled at N equally spaced frequencies between 0 and 2p:

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N

k

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nx --

=å=1

0

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n

WnxkX å-

=

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~~

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k

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[ ]kX [ ]kX~[ ]nx~[ ]nx

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𝑥"[𝑛] 𝑋'[𝑘] ={𝑁𝑎,}= 𝑥"[𝑛]DFS/IDFS IDTFS/DTFS

𝑋 𝑘 = 𝑋 𝑒01 ×S𝑘𝛿 𝑛 − 2p𝑘/𝑁𝑥"[n]=𝑥 𝑛 ∗ ∑ 𝛿 𝑛 − 𝑘𝑁�,

DFT/IDFT as Matrix Operation(ppt slides only)

l DFT

l Inverse DFT

l Computational Complexityl Computation of an N-point DFT or inverse

DFT requires N 2 complex multiplications.

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1

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Properties of the DFS/DFT

Circular Time/Frequency Shiftl Circular Time Shift (proved by DFS property of )

l Circular Frequency Shift (IDFS property of )

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NN

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=«-

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Njln

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Circular Convolutionl Defined for two N-length sequences as

l Circular convolution in time is multiplication in frequency

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[ ] [ ] [ ] [ ]kXkXnxnx N 2121 !«×Duality

Computing Circular Convolution;Circular vs. Linear Convolution

l Computing circular convolution:l Linearly convolve and

l Place sequences on circle in opposite directions, sum up all pairs, rotate outer sequence clockwise each time increment

l Circular versus Linear Convolution

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n0 3

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1 2

[ ] [ ]nxnx 21 4

n0 3

4

1 2

x1[n] * x2[n]

n0 3

1

1 2 64 5

1

2 2

3 3

4 Can obtain a linear convolution from a circular one by zero

padding both sequences

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Main Points

l DFS/DFT have similar properties as DTFS/DTFT but with modifications due to periodic/circular characteristics

l A circular time shift leads to multiplication in frequency by a complex phase term

l A circular frequency shift leads to a complex phase term multiplication with the original sequence (modulation)

l Circular convolution in time leads to multiplication of DFTs

l Circular convolution can be computed based on linear convolution of periodized sequences or circle method