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Discrete-Time Fourier Transform (DTFT) and Discrete Fourier Transform (DFT)

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Learn about computing DTFT, difference between DTFT and DFT, inverse DTFT, DFT limitations, and Fast Fourier Transform (FFT) for efficient signal processing. Dive into signal flow diagram, Z-transform, and power spectrum density concepts in signal processing.

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Discrete-Time Fourier Transform (DTFT) and Discrete Fourier Transform (DFT)

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  1. 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

  2. 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

  3. 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

  4. 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?

  5. To recover x(k) from X(f) • Use inverse DTFT

  6. Some characteristics • X(f+fs) = X(f) • X(-f) = X*(f) • It is also common to represent DTFT in normalized frequency:

  7. 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.

  8. Introducing roots of unity Where WNk = cos(2pk/N)-jsin(2pk/N)

  9. To get X(i) from x(k) 0 ≤ i < N

  10. 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

  11. Example • Order of DFT = N2 Because is a N by N matrix multiplication

  12. Signal spectrum • Recall from continuous domain • Fourier coefficients ci can be obtained from the DFT of the sampled xa(t) • Power spectrum density (PSD):

  13. 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

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