G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Continuous (or analogue) signals Continuous signals (Single or multiple) t y(t) u(t) Discontinuity G(s) u(t)U(s) y(t) Y(s)
G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform The sampling process
The definition and the properties
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 The z-transform • Classification of signals
• Sampling of continuous signals
• The z-transform: definition
• The z-transform: properties
• Inverse z-transform
• Application to systems
• Comments on stability
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Continuous (or analogue) signals Classification of signals • Sampled signals • Discrete (or digital or time) signals
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Continuous (or analogue) signals Continuous signals (Single or multiple) t y(t) u(t) Discontinuity G(s) u(t)U(s) y(t) Y(s)
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 x(t) t x[kT] t Amplitude
modulated t x[kT] Pulsewidth
modulated Original
signal Sampled signal: modulation techniques
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Noted x’(t)
• Samples exist only at sampling times
• The relative height represents the value (information)
• The sampling T period is constant
• Able to drive a physical system t x’(t) T kT (K+1)T Sampled signal
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Physical sample t h Information Area=h Dirac (or unit) pulse
As
(Distribution) NB: The energy of a sample pulse is finite
(able to drive a physical system)
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Noted x*(t)
• Values defined only at sampling times
• The relative height represents the numerical value
• The sampling T period is constant
• Usable for arithmetic operations
• Unable to drive a physical system t x*(t) T kT (K+1)T Discrete (or digital or time) signals
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 •Application G(s) u(t) y(t) Continuous signal Measure
(ADC) Driver
(DAC) Computer Discrete signal Sampling Samples or steps
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Continuous system G(s) x(t) x(t) t y(t) t y(t)
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Sampled system x’(t) t y(t) t G(s) x’(t) y(t)
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Sampled system with hold circuit G(s) x’(t) y(t) u(t) t y(t) t Hold x’(t) t u(t)
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 • Discrete system D(z) x*(t) y*(t) y*(t) t x*(t) t D(z) defined by difference equations or by transfer function
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 The z-transform • Classification of signals
• Sampling of continuous signals
• The z-transform: definition
• The z-transform: properties
• Inverse z-transform
• Application to systems
• Comments on stability
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Sampling time and delay t t=kT t=(k-1)T t=(k+1)T t=(k+)T x[k-1] x[k] x[k+1] x[k,] x(kT)=x[k,]=x[k]=x k Several notations
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 t Sampling times (k-1)T (k+1)T kT Sampling function n=integer………-2,-1,0,1,2,3,4,……….. Periodic function
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Decomposition in Fourier series n=0 n=1 n=-1 0
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Ideal sampler t f(t) t
G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Sampling process Continuous
Function
f(t) Sampler
(t) Series of
samples T Reconstruction ?
Comments