Lecture 15 Outline: DFT Properties and Circular Convolution Announcements: HW 4 posted, due Tues May 8 at 4:30pm. No late HWs as solutions will be available immediately. Midterm details on next page HW 5 will be posted Fri May 11, due following Fri (as usual) Review of Last Lecture DFT as a Matrix Operation Properties of DFS and DFT Circular Time/Freq. Shift and Convolution Circular Convolution Methods Linear vs. Circular Convolution

Midterm Details Time/Location: Friday, May 11, 1:30-2:50pm in this room. 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 1-13. Practice MT posted, worth 25 extra credit points for “taking” it. 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) Solutions given when you turn in your answers In addition to practice MT, we will also provide additional practice problems/solns MT Review in class May 7 Discussion Section May 8, 4:30-6 (MT review and practice problems) Regular OHs for me/TAs this week and next (no new HW next week) I am also available by appointment

Review of Last Lecture Discrete Fourier Series (DFS) Pair for Periodic Signals Discrete Fourier Transform (DFT) Pair and are one period of and , respectively DFT is DTFT sampled at N equally spaced frequencies between 0 and 2p: DFS/IDFS IDTFS/DTFS 𝑥 [𝑛] 𝑋 [𝑘] ={𝑁 𝑎 𝑘 }= 𝑥 [𝑛] 𝑋 𝑘 =𝑋 𝑒 𝑗Ω 𝑘𝛿 𝑛−2𝑘/𝑁 𝑥 [n]=𝑥 𝑛 ∗ 𝑘 𝛿 𝑛−𝑘𝑁

DFT/IDFT as Matrix Operation (ppt slides only) Inverse DFT Computational Complexity Computation of an N-point DFT or inverse DFT requires N 2 complex multiplications.

Properties of the DFS/DFT

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

Circular Convolution Defined for two N-length sequences as Circular convolution in time is multiplication in frequency Duality

Computing Circular Convolution; Circular vs. Linear Convolution Linearly convolve and Place sequences on circle in opposite directions, sum up all pairs, rotate outer sequence clockwise each time increment Circular versus Linear Convolution x1[n] * x2[n] Can obtain a linear convolution from a circular one by zero padding both sequences

Main Points DFS/DFT have similar properties as DTFS/DTFT but with modifications due to periodic/circular characteristics A circular time shift leads to multiplication in frequency by a complex phase term A circular frequency shift leads to a complex phase term multiplication with the original sequence (modulation) Circular convolution in time leads to multiplication of DFTs Circular convolution can be computed based on linear convolution of periodized sequences or circle method