Online Lectures

Note: The videos below are on YouTube. I recommend setting the playback speed of the videos to 1.25 or 1.5 times real-time.

Discrete Fourier Transform

1 (in class)

• Definition: analysis & synthesis formulae
• Definition: $W_N = e^{-j\frac{2\pi kn}{N}}$
• Transform between two $N$-point sequences

2 (video)

• Matrix-vector formulation of DFT and inverse
• Matlab dftmtx (DFT matrix) function
• Look at image of real and imaginary parts for $N=64$
• Look at time plots of real part for $N=64$. The DFT matrix contains all the sinusoids at all harmonic frequencies $f=k/N, k=0, 1, \cdots, N-1$.

3 (video 1) (video 2)

• Relation between DFT and DTFT
• Relation between DFT and CTFT (no aliasing)
• Frequency domain sampling and time-domain aliasing. Why the length of the sequence $L$ should be less than the length of the transform $N$.
• Zero padding is an application of the DFT-DTFT relation
• What you get from zero padding versus using more data (i.e. longer window)

4 (video)

• Derive the DFT of a complex exponential sequence.
• Discuss $N$-point DFT bin center frequencies: $f=k_0/N, k_0=0,1,2,\cdots, N-1$ and the sequence length $L=N$, i.e. no zero padding. If these two conditions are satisfied, then $X[k] = N\delta[k-k_0]$.
• Discuss how zero padding leads to more samples of the same underlying function. These samples help humans to better visualize the function, but they do not bring in any new information.

5 (video)

• Plotting DFTs in Matlab

6 (video)

• Modulo property of $e^{j\frac{2\pi kn}{N}}$
• Circular time and frequency shift properties
• What a circular shift looks like

7 (multiplication) (convolution)

• Circular convolution property
• Multiplication property

8 (video)

• Windowing property of the DTFT and DFT

9 (video)

• Analysis and synthesis formulas of DFT
• 1D and 2D visualizations in Matlab

10 (video)

• Circular time reversal property of the DFT

11 (video)

• Circular convolution property of the DFT
• Illustration of circular convolution in Matlab
• How to make circular convolution perform linear convolution by zero padding

12 (video)

• Windows and windowing
• Users choose window length and shape
• Length and shape affect main lobe width which determines resolution
• Shape affects side lobe level which determines spectral leakage
• Matlab examples are given
• Slides

Z-Transform

The z-transform is discussed in chapter 3 in our textbook.

1 (video) Part 1 (slides)

2 (video) Part 2

System Analysis using the Z-Transform

These topics are discussed in chapter 5 in our textbook.

1 (video) (slides)

Filter Design

1 (video) (slides1) (slides2)

2 (HW 10 video) (work)