Lecture 16 Outline: Discrete Fourier Series and Transforms Announcements: Reading: “5: The Discrete Fourier Transform” pp 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. 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 (two floors up) We should be aware of all conflicts with the 9:20-11:20am timeslot at this time 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 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 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)

Review of Last Wed 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 h d [n] entails picking a length M, setting h a [n]=h d [n], |n|  M/2, choosing window w[n] with h w [n]=h[n]w[n]to mitigate Gibbs, and setting h[n]=h w [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 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 DFT Inverse DFT DFT/IDFT commonly used in DSP, using N-length signal blocks, due to its much lower computational complexity than the DTFT/IDTFT Conjugate Relationships

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

Relation between DFT and DTFT 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 2  : or

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

Properties of the DFS/DFT

Properties (Continued)

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 (N 2 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