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Things to do. Proposal / Human subject consent form Solution to assignment 2 will be up at 1:05pm Feedback on labs Test 1 this Thursday (up to Chapter 2) Sampling, aliasing and quantization Transfer function, difference equation, zero-pole, signal flow diagram
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Things to do • Proposal / Human subject consent form • Solution to assignment 2 will be up at 1:05pm • Feedback on labs • Test 1 this Thursday (up to Chapter 2) • Sampling, aliasing and quantization • Transfer function, difference equation, zero-pole, signal flow diagram • Z-transform (system function / impulse response) • Inverse Z transform (residue, partial fraction) • General pre-filter stage design
BIEN425 – Lecture 6 • By the end of the lecture, you should be able to: • Compute the discrete-time Fourier transform (DTFT) of a signal • Describe the difference between DTFT and discrete Fourier transform (DFT) • Compute the DFT of a discrete time signal • Compute the power spectrum density of a signal
Recall that the Fourier transform(FT) converts a continuous time-domain function to a continuous frequencydomainfunction. • Now, given the Z-transform of a discrete-time signal X(z) • Also note that X(f) is complex
Discrete-time Fourier transform • Definition of DTFT: • In order for the series x(k) to converge, all poles of X(z) must be within the unit circle Remember partial fraction expansion? Infinite series and convergence?
To recover x(k) from X(f) • Use inverse DTFT
Some characteristics • X(f+fs) = X(f) • X(-f) = X*(f) • It is also common to represent DTFT in normalized frequency:
Discrete Fourier transform (DFT) • Limitation of DTFT in practice: • Infinite number of arithmetic operations • Infinite number of points in creating f-domain • Solution: Evaluate at N distinct frequencies fi=fs/N, where 0≤ i<N • N determines 1) how many input values are needed 2) Resolution of freq domain results. 3) Processing time required for DFT.
Introducing roots of unity Where WNk = cos(2pk/N)-jsin(2pk/N)
To get X(i) from x(k) 0 ≤ i < N
If we write x(k) as a column matrix, W as a square matrix, we can get X(i) very easily. • DFT (Analysis equation) • Inverse DFT (Synthesis equation) • Let’s recap our notations
Example • Order of DFT = N2 Because is a N by N matrix multiplication
Signal spectrum • Recall from continuous domain • Fourier coefficients ci can be obtained from the DFT of the sampled xa(t) • Power spectrum density (PSD):
As you can see DFT is not very affordable (N2), we will introduce a faster and more effective way of computing DFT, called Fast Fourier Transform (FFT) • Let’s check out some examples of the FFT function using Matlab (fft, fftshift) in more details