1. 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

2. 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

3. 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

4. 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 .

5. 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

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

7. Relation between DFT and DTFTor
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:

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

9. Properties of the DFS/DFT

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

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

12. 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