Lecture 14 Outline: Discrete Fourier Series and Transforms 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 Discrete Fourier Series Discrete Fourier Transform Relation between DFT & DTFT DFT as a Matrix Operation Properties of DFS and DFT Circular Time/Freq. Shift and 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 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 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 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 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 FIR implemented directly using M delay elements and M+1 multipliers Can introduce group delay Efficiently implemented with DFT Example design for LFP (Differentiator in HW) 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 : Then is also N-periodic: Define Appears in the DTFS, DFS or DFT for N-periodic sequences Then we can write Using this notation, we have the DFS pair for periodic signals: Simple computation of makes this pair easier to compute than DTFS . Usin ≜ g this notation, we write as .

Discrete Fourier Transform (DFT) Works with only one period of and Can recover original periodic sequences , as Equivalently, work with N samples of x[n] Leads to DFT Pair Conjugate Relationships Inverse DFT DFT DFT/IDFT commonly used in DSP, using N-length signal blocks, due to its much lower computational complexity than the DTFT/IDTFT

Example Real and even and One-period representations: . . . . . .

Relation between DFT and DTFT or Given a length-N signal x[n], n-point DFT is Its DTFT is DFT is the DTFT sampled at N equally spaced frequencies between 0 and 2p:

DFT/IDFT as Matrix Operation 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

Main Points DFS is the DTFS with a different normalization DFT operates on one N-length “piece” of a signal x[n] Fast/low complexity computation (N2 complex multiplications) DFT is the DTFT sampled at N equally spaced frequencies between 0 and 2p 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