ELEN 5346/4304 DSP and Filter Design Fall Lecture 4: Frequency domain representation, DTFT, IDTFT, DFT, IDFT Instructor: Dr. Gleb V. Tcheslavski Contact: Office Hours: Room 2030 Class web site: ex.htm ex.htm

ELEN 5346/4304 DSP and Filter Design Fall Some history Jean Baptiste Joseph Fourier was born in France in He attended the Ecole Royale Militaire and in 1790 became a teacher there. Fourier continued his studies at the Ecole Normale in Paris, having as his teachers Lagrange, Laplace, and Monge. Later on, he, together with Monge and Malus, joined Napoleon as scientific advisors to his expedition to Egypt where Fourier established the Cairo Institute. In 1822 Fourier has published his most famous work: The Analytical Theory of Heat. Fourier showed how the conduction of heat in solid bodies may be analyzed in terms of infinite mathematical series now called by his name, the Fourier series.

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation frequency complex exponent of  0 complex exponent of -  0 A sinusoidal signal is represented by TWO complex exponents of opposite frequencies in the frequency domain. (4.3.1) (4.3.2)

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation (cont) Iff it exists! has a period of 2  (4.4.1) (4.4.2) (4.4.3) (4.4.4)

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation (cont 2) For an arbitrary real LTI system: Symmetric with respect to  Anti-symmetric with respect to 

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation (cont 3) Combining (4.3.2) and (4.4.4) – back to our sinusoid! LTI filtering: due to the input change due to the system same as the input from the input phase change due to the system Via design, we manipulate H(e j  ), therefore, h n, and, finally, manipulate the coefficients in the Linear Constant Coefficient Difference Equation (LCCDE) (4.6.1) (4.6.2) (4.6.3) (4.6.4)

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation (cont 4) LCCDE: for large enough n: (4.7.1) (4.7.4) (4.7.2) (4.7.3)

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation (cont 5) for an LTI: We don’t need systems of order higher than 2: can always make cascades. for a real, LTI, BIBO system: effects of filtering We cannot observe ANY frequency components in the output that are not present in the input (in steady state). We may see less when (4.8.1) (4.8.2)

ELEN 5346/4304 DSP and Filter Design Fall Frequency domain representation (cont 6) In continuous time: signalnoiseconstdelay We need a constant magnitude and linear phase for the frequencies of interest. Ideal filters:     LPF   HPF   BPF   BSF Ideal filters are non-realizable! (4.9.1) (4.9.2)

ELEN 5346/4304 DSP and Filter Design Fall CTFT and ICTFT (4.10.1) (4.10.2) CTFT: ICTFT: Examples:

ELEN 5346/4304 DSP and Filter Design Fall DTFT if exists (4.11.1) What’s about convergence??? 1. Absolute convergence: (4.11.4) (4.11.5) (4.11.2) (4.11.3)

ELEN 5346/4304 DSP and Filter Design Fall DTFT (cont) must be 2. Mean-square convergence: The total energy of the error must approach zero, not an error itself! (4.12.1) Absolutely summable sequences always have finite energy. However, finite energy sequences are not necessary absolutely summable.

ELEN 5346/4304 DSP and Filter Design Fall IDTFT IDTFT: (4.13.1) (4.13.2) Combining (4.11.1) and (4.12.2) (4.13.3) (4.13.4) shows where x n “lives” in the frequency domain.

ELEN 5346/4304 DSP and Filter Design Fall Back to ideal filters Ideal LPF:   22 0 1 cc Using IDTFT: 1.The response in (4.14.2) is not absolutely summable, therefore, the filter is not BIBO stable! 2.The response in (4.14.2) is not causal and is of an infinite length. (4.14.1) (4.14.2) As a result, the filter in (4.14.1) is not realizable. Similar derivations show that none of the ideal filters in slide 9 is realizable.

ELEN 5346/4304 DSP and Filter Design Fall DTFT properties (4.15.1) (4.15.2) (4.15.3) (4.15.4) (4.15.5) (4.15.6) continuous, periodic functions (4.15.6)

ELEN 5346/4304 DSP and Filter Design Fall DTFTs of commonly used sequences

ELEN 5346/4304 DSP and Filter Design Fall DTFT examples ½ of DC value of u n (4.17.1) (4.17.2) (4.17.3) (4.17.4) (4.17.5) (4.17.6)

ELEN 5346/4304 DSP and Filter Design Fall DTFT examples (cont) We can re-work the Parseval’s theorem (4.15.6) as follows: energy density (spectrum) Autocorrelation function: (4.18.1) (4.18.2)

ELEN 5346/4304 DSP and Filter Design Fall DTFT examples (cont 2) One obvious problem with DTFT is that we can never compute it since x n needs to be known everywhere! which is impossible! Therefore, DTFT is not practical to compute. Often, a finite dimension LTI system is described by LCCDE: (4.19.2) practical (finite dimensions) Prediction of steady-state behavior of LCCDE (4.19.1)

ELEN 5346/4304 DSP and Filter Design Fall How to measure frequency response of an actual (unknown) filter? 1. Perform two I/O experiments: 2. Analyze these measurements and form: That’s a good way to measure/estimate a frequency response for every . (4.20.1) (4.20.2) (4.20.3) (4.20.4) (4.20.5)

ELEN 5346/4304 DSP and Filter Design Fall DFT and IDFT Consider an N-sequence x n (at most N non-zero values for 0  n  N-1) uniformly spaced frequency samples (4.21.1) (4.21.2) DFT: Finite sum! Therefore, it’s computable. (4.21.1) can be rewritten as: (4.21.3) (4.21.4) (4.21.5) Btw, DFT is a sampled version of DTFT.

ELEN 5346/4304 DSP and Filter Design Fall DFT and IDFT (cont) Let us verify (4.21.5). We multiply both sides by (4.22.1) (4.22.2) (4.22.3) (4.22.4)

ELEN 5346/4304 DSP and Filter Design Fall FFT In the matrix form: where: (4.23.1) (4.23.2) (4.23.3) (4.23.4) (4.23.5) (4.23.6) (4.23.7) This is actually FFT…

ELEN 5346/4304 DSP and Filter Design Fall Relation between DTFT and DFT 1. Sampling of DTFT (4.24.1) (4.24.2) (4.24.3) y n is an infinite sum of shifted replicas of x n. Iff x n is a length M sequence (M  N) than y n = x n. Otherwise, time-domain aliasing  x n cannot be recovered!

ELEN 5346/4304 DSP and Filter Design Fall Relation between DTFT and DFT (cont) 2. DTFT from DFT by Interpolation Let x n be a length N sequence: It’s possible to determine DTFT X(e j  ) from its uniformly sampled version uniquely! Let us try to recover DTFT from DFT (its sampled version). (4.25.1) (4.25.2) (4.25.3)

ELEN 5346/4304 DSP and Filter Design Fall Relation between DTFT and DFT (cont 2) 3. Numerical computation of DTFT from DFT Let x n is a length N sequence: defined by N uniformly spaced samples We wish to evaluate at more dense frequency scale. No change in information, no change in DTFT… just a better “plot resolution”. (4.26.1) Define:zero-padding (4.26.2) (4.26.3)

ELEN 5346/4304 DSP and Filter Design Fall A note on W N W N is also called an N th root of unity, since Re Im Re Im (4.27.1) (4.27.2)

ELEN 5346/4304 DSP and Filter Design Fall DFT properties 1. Circular shift x n is a length N sequence defined for n = 0,1,…N-1. An arbitrary shift applied to x n will knock it out of the 0…N-1 range. Therefore, a circular shift that always keeps the shifted sequence in the range 0…N-1 is defined using a modulo operation: (4.28.1) (4.28.2)

ELEN 5346/4304 DSP and Filter Design Fall DFT properties (cont) 2. Circular convolution A linear convolution for two length N sequences x n and g n has a length 2N-1: A circular convolution is a length-N sequence defined as: N N (4.29.1) (4.29.2) (4.29.3) Procedure: take two sequences of the same length (zero-pad if needed), DFT of them, multiply, IDFT: a circular convolution.

ELEN 5346/4304 DSP and Filter Design Fall DFT properties (cont 2) Example: Take N frequency samples of (4.30.1) and then IDFT: aliased version of x n (4.30.1) (4.30.2) The results of circular convolution differ from the linear convolution “on the edges” – caused by aliasing. To avoid aliasing, we need to use zero-padding…

ELEN 5346/4304 DSP and Filter Design Fall Linear filtering via DFT Often, we need to process long data sequences; therefore, the input must be segmented to fixed-size blocks prior LTI filtering. Successive blocks are processed one at a time and the output blocks are fitted together… Assuming that h n is an M-sequence, we form an N-sequence (L - block length): (4.31.1) We can do it by FFT: IFFT{FFT{x}FFT{h}}… N >> M; L >> M; N = L + M - 1 and is a power of 2 Problem: DFT implies circular convolution – aliasing! (4.31.2)

ELEN 5346/4304 DSP and Filter Design Fall Linear filtering via DFT (cont) Next, we compute N-point DFTs of x m,n and h n, and form (4.32.1) - no aliasing! Since each data block was terminated with M -1 zeros, the last M -1 samples from each block must be overlapped and added to first M – 1 samples of the succeeded block. An Overlap-Add method.

ELEN 5346/4304 DSP and Filter Design Fall Linear filtering via DFT (cont 2) Alternatively: Each input data block contains M -1 samples from the previous block followed by L new data samples; multiply the N- DFT of the filter’s impulse response and the N-DFT of the input block, take IDFT. Keep only the last L data samples from each output block. An Overlap-Save method. The first block is padded by M-1 zeros.

ELEN 5346/4304 DSP and Filter Design Fall DFT properties: General from Mitra’s book Btw, g[n] = g n

ELEN 5346/4304 DSP and Filter Design Fall DFT properties: Symmetry from Mitra’s book x n is a real sequence

ELEN 5346/4304 DSP and Filter Design Fall DFT properties: Symmetry (cont) from Mitra’s book x n is a complex sequence

ELEN 5346/4304 DSP and Filter Design Fall N-point DFTs of 2 real sequences via a single N-point DFT Let g n and h n are two length N real sequences. Form x n = g n + jh n  X k (4.37.1) (4.37.2) (4.37.3) (4.37.4)

ELEN 5346/4304 DSP and Filter Design Fall Summary AlgorithmTimeFrequency CTFTContinuous DTFTDiscreteContinuous DFTDiscrete