#### Transcript 06filled

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 1 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 2 • Recall that the Fourier transform (FT) converts a continuous time-domain function to a continuous frequency domain function. • Now, given the Z-transform of a discrete-time signal X(z) X ( f ) X ( z ) z e j 2 fT • Also note that X(f) is complex X ( f ) A( f )e j ( f ) 3 Discrete-time Fourier transform • Definition of DTFT: X(f) x (k )e j 2 kfT k 0 • 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? 4 To recover x(k) from X(f) • Use inverse DTFT fs / 2 X ( f )e fs / 2 fs / 2 ji 2 fT jk 2 fT df x ( k ) e fs / 2 k 0 fs / 2 e x(k ) k 0 x(k ) e k 0 df x(k ) j ( i k ) f sT e j ( i k ) f sT j 2 T ( i k ) k 0 j ( i k ) 2 fT fs / 2 ji 2 fT df e 1 sin( i k ) T ( i k ) x ( k ) f (i k ) x (i ) f s k 0 s 5 Some characteristics • X(f+fs) = X(f) • X(-f) = X*(f) A( f ) A( f ) ( f ) ( f ) even odd • It is also common to represent DTFT in normalized frequency: fˆ f s / 2 0 . 5 fˆ 0 . 5 6 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 1 X(f) x (k )e j 2 fkT k 0 • N determines 1) how many input values are needed 2) Resolution of freq domain results. 3) Processing time required for DFT. 7 Introducing roots of unity WN e j 2 N Where WNk = cos(2k/N)-jsin(2k/N) 8 To get X(i) from x(k) X (i ) X ( f ) f f i if s / N N 1 X (i ) x (k )e 0≤i<N j 2 ( if s / N ) kT k 0 N 1 X (i ) x (k )e j ( 2 k / N ) i k 0 N 1 X (i ) x ( k )W ik N k 0 9 • DFT (Analysis equation) If we write x(k) as a column matrix, W as a square matrix, we can get X(i) very easily. X Wx • Inverse DFT (Synthesis equation) x 1 N * W X • Let’s recap our notations x(t) t: real x(k) k: integer x: continuous; time-domain; real function x: discrete-time; time-domain; real function X(z) z: complex X: Z-transform; z-domain; complex function X(f) f: real X: DTFT; frequency-domain; complex function X(i) i: interger X: DFT; frequency-domain ;complex function 10 Example 3 1 x 0 2 ( ( W ( ( j) 0 ( j) 0 ( j) 0 j) 0 ( j) 1 ( j) 2 j) 0 ( j) 2 ( j) 4 j) 0 ( j) 3 ( j) 6 ( j) 3 ( j) 6 ( j) 9 ( j) 0 4 3 j 3 X Wx 2 3 j 3 • Order of DFT = N2 Because is a N by N matrix multiplication 11 Signal spectrum • Recall from continuous domain 1 j 2 if 0 t ci x a (t ) e dt 0 • Fourier coefficients ci can be obtained from the DFT of the sampled xa(t) ci 1 N 1 x (k )e N j 2 ik / N X (i ) k 0 • Power spectrum density (PSD): N S N (i ) X (i ) N 2 12 Full name Abbrev. Time Freq Fourier Transform FT C C Discrete-time Fourier Transform DTFT D C Discrete Fourier Transform DFT D D • 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 13