1 Discrete Hilbert Transform 7 th April 2007 Digital Signal Processing I Islamic University of Gaza

2 Overview  Hilbert Transforms  Discrete Hilbert Transforms  DHT in Periodic/Finite Length sequences  DHT in Band pass Sampling

3 Transforms  Laplace Transforms Time domain  s-plane  Fourier Transforms (FT/DTFT/DFT) Time domain  frequency domain  Z- Transforms Time domain  Z domain ( delay domain )  Hilbert Transforms For Causal sequences relates the Real Part of FT to the Imaginary Part FT

4 Why Hilbert Transforms ?  Fourier Transforms require complete knowledge of both Real and Imaginary parts of the magnitude and phase for all frequencies in the range – π < ω < π  Hilbert Transforms applied to causal signals takes advantage of the fact that Real sequences have Symmetric Fourier transforms.

5 Because of the possible singularity at x=t, the integral is considered as a Cauchy Principal value Analog Hilbert Transforms The Hilbert Transform of the function g(t) is defined as The forms of the Hilbert Transform are So the Hilbert transform is a Convolution

6 A note on Symmetry  For real signals we have the following Fourier transforms relationships Any complex signal can be decomposed into parts having Conjugate Symmetry ( even for real signals) Conjugate Anti-Symmetry (odd for real signals) (1) (2) (3) (4) (5) (6)

7 A note on Symmetry … x[n] x e [n] x o [n] x[-n] n

8 Problem1 1) Find Xi(w)

9 Problem2 2) Find X(z)

10 Derivation of Hilbert Transform Relationships

11 The Hilbert Transform Relationships The above equations are called discrete Hilbert Transform Relationships hold for real and imaginary parts of the Fourier transform of a causal stable real sequence. Where P is Cauchy principle value

12  Note:  A periodic sequence cannot be casual in the sense used before, but we will define a “ periodically causal ” sequence  Henceforth we assume N is even Definitions: Periodic Sequences

13 Periodic Sequences …

14 x ~ [n] x ~ [-n] n x ~ o [n] x ~ e [n] Periodic Sequences …

15 1.Compute x ~ e [n] from X ~ R [k] using DFS synthesis equation 2.Compute x ~ [n] from x ~ e [n] 3.Compute X ~ [k] from x ~ [n] using DFS analysis equation Periodic Sequences …

16 Finite Length Sequences  It is possible to apply the transformations derived if we can visualize a finite length sequence as one period of a periodic sequence.  For all time domain equations replace x ~ (n) with x(n)  For freq domain equations ---

17 Problem 3  N=4, X R [k]=[ ], Find X I [k]  Method 1  V 4 [k]=[ … 0 -2j 0 2j … ]  jX I [k]=[ 0 j 0 – j ]  Method 2  x e [n]=[ 3 -1/2 0 -1/2 ]  x o [n]=[ 0 -1/2 0 -1/2 ]  jX I [k]=[ 0 j 0 – j ]

18 Relationships between Magnitude and Phase  We obtain a relationship between Magnitude and phase by imposing causality on a sequence x^(n) derived from x(n) The fact that the minimum phase condition ( X(z) has all poles and zeros inside the unit circle) guarantees causality of the complex cepstrum.

19 Complex Sequences  Useful in useful in representation of bandpass signals  Fourier transform is zero in 2 nd half of each period.  Z-Transform is zero on the bottom half  The signal called an analytic signal (as in continuous time signal theory)

20 Complex Sequences … Note: Such a system is also called a 90º phase shifter. -x r [n] can also be obtained form a x i [n] using a 90º phase shifter

21 Complex Sequences … Hilbert Transformer X r [n] X i [n]

22 Representation of Bandpass Signals

23 Representation of Bandpass Signals … s r [n] sin(w c n) x r [n] s i [n] Hilbert Transformer X + X cos(w c n) Hilbert Transformer + +- Hilbert Transformer X X + sin(w c n) cos(w c n) x r [n] +

24 Bandpass Sampling C/D Hilbert Transformer ↓M T S r [n]=S c [nT] S id [n] S rd [n] S i [n] Sc(t)

25 Bandpass Sampling … Reconstruction of the real bandpass signal involves 1.Expand the complex signal by a factor M 2.Filter the signal using an ideal bandpass filter 3.Obtain S r [n]=Re{s e [n]*h[n]}

26 Concluding Remarks  Relations between Real and Imaginary part of Fourier transforms for causal signal were investigated  Hilbert transform relations for periodic sequences that satisfy a modified causality constraint  When minimum phase condition is satisfied logarithm of magnitude and the phase of the Fourier transform are a Hilbert transform pair  Application of complex analytic signals to the efficient sampling of bandpass signals were discussed

27 References  Discrete Time Signal Processing, 2 nd Edition. © 1999 Chapter 11 pages , Alan V Oppenheim, Ronald W Schafer with John R Buck