Signals and Sysems Lecure 1-3 1
Cascading Sysem ( ) 2 Audio & DSP Lab.
Discree-ime-shif operaor S k, operaing on he discree-ime signal x[n] o produce x[n k]. ( S k ) 3
Two implemenaions of he moving-average sysem (a) cascade form of implemenaion and (b) parallel form of implemenaion. 4
Properies of Sysems Sabiliy Memory Causaliy Inveribiliy Time Invariance Lineariy 5
Sabiliy Bounded Inpu causes Bounded Oupu (BIBO). y( ) M x( ) M for M x y x, M y < < are finie and posiive numbers. Unsable Example: Tacoma Narrows Suspension Bridge 6
Dramaic phoographs showing he collapse of he Tacoma Narrows suspension bridge on November 7, 1940. (a) Phoograph showing he wising moion of he bridge s cener span jus before failure. (b) A few minues afer he firs piece of concree fell, his second phoograph shows a 600-f secion of he bridge breaking ou of he suspension span and urning upside down as i crashed in Puge Sound, Washingon. Noe he car in he op righ-hand corner of he phoograph. (Couresy of he Smihsonian Insiuion.) 7
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10 Audio & DSP Lab. Example of a Sable Sysem ( ). 2] [ 1] [ ] [ 3 1 ] [ : + + n x n x n x n y Example [ Moving-average case ] ( ) ( ) ( ) x x x x M M M M n x n x n x n x n x n x n y + + + + + + 3 1 2] [ 1] [ ] [ 3 1 2] [ 1] [ ] [ 3 1 ] [
Example of an Unsable Sysem Example : y[ n] n r x[ n], r > 1 even if x[ n] M x <, all n y[ n] r n x[ n] r n x[ n] if r > 1, r n divergences 11
Memory : Memory or Memory-less Memory-less : 12
Causaliy Causal : Ex : y[ n] x[ n] + x[ n 1] Non-causal : Ex : y[ n] x[ n + 1] + x[ n] 13
Example: Causal Sysem? Series RC circui driven from an ideal volage source v 1 (), producing oupu volage v 2 () 14
Sysem Inveribiliy The second operaor H inv is he inverse of he firs operaor H. Hence, he inpu x() is passed hrough he cascade correcion of H and H -1 compleely unchanged. H inv H 1 15
Time Invariance These wo siuaions are equivalen, provided ha H is ime invarian. y ( ) y1( 0) 2 16
The Lineariy Properies If hese wo configuraions produce he same oupu y(), he operaor H is linear. N i 1 a x ( ) i i 17
EX: RC circui : y()? in response o he uni-impulse inpu x() δ(). sep response : y( ) ( / RC 1 e ) u( ) ( y() ) 18
19 Audio & DSP Lab. Recangular pulse of uni area approaches a uni impulse as 0 ) ( lim ) ( 0 x δ x () ) ( ) ( 2 1 2 1 ) ( 2 1 x x u u x +
20 Audio & DSP Lab. Sysem Oupu y(), applying Linear propery: )} ( ) ( { ) ( ) ( ) ( )}; ( { ) ( )}; ( { ) ( 2 1 2 1 2 2 1 1 x x H y y y x H y x H y ± ± ( ) ) ( 1 ) ( ) ( ) ( ) 2 ( 1 1 ) ( ) 2 ( 1 1 ) ( : / 2 1 )/(RC) 2 ( 2 )/(RC) 2 ( 1 u e RC y y y u e y u e y soluion RC + + ( )
EX: RL circui : y() in response o he uni-impulse inpu x() δ()? 21
Waveform of elecrical noise generaed by a hermionic diode wih a heaed cahode. Noe ha he ime-averaged value of he noise volage displayed is approximaely zero. 22
(a) Thévenin equivalen circui of a noisy resisor. (b) Noron equivalen circui of he same resisor. 23
Simple RC circui wih small ime consan, used as an approximaor o a differeniaor. 24
Simple RC circui wih large ime consan used as an approximaor o an inegraor. 25
Mechanical lumped model of an acceleromeer 2 d y( ) 2 d + D M dy( ) d + K M y( ) x( ) 26
Radar Range Measuremen d τ C τ 2d C 27
Radar Range Measuremen (con.) (Range Resoluion) T 0 : d min ct 2 0 meers (Range Ambiguiy) T : d max ct 2 meers 28
Moving Average Sysem Flucuaions in he closing sock price of Inel over a hree-year period. 29
Oupu of a 4-poin moving-average sysem 30
Oupu of an 8-poin moving-average sysem 31
Muliple propagaion pahs in a wireless communicaion environmen 32
Tapped-delay-line model of a linear communicaion channel, assumed o be ime-invarian 33
Block diagram of firs-order recursive discree-ime filer The operaor S shifs he oupu signal y[n] by one sampling inerval, producing y[n 1]. The feedback coefficien p deermines he sabiliy of he filer. 34
Exponenially damped sinusoidal sequence 35
Exploring Conceps wih MATLAB Periodic Signals Exponenial Signals Sinusoidal Signals Exponenially Damped Sinusoidal Signals Sep, Impulse and Ramp Funcions User Defined Funcions 36
MATLAB Window 37 Audio & DSP Lab.
Periodic Signal Generae Square Wave: A 1; w0 10* pi; rho 0.5; 0 : 0.001:1; sq A* square( w0*, rho); plo(, sq) axis([0 1 1.1 1.1]) 38
Exponenial Signal Generae Funcion: x b exp(-a) b 5; a 6; 0 : 0.001:1; x b*exp( a* ); plo(, x) 39
Exp Signal Plo 40 Audio & DSP Lab.
Sinusoidal Signal Generae Funcion: x A cos(ω o + φ) A 4; w0 20* pi; phi pi / 6; 0 : 0.001:1; x A*cos( w0* + phi); plo(, x) 41
Sine Signal Plo 42 Audio & DSP Lab.
Exponenially Damped Sinusoidal Signal Generae Funcion: x A e -a sin(ω o + φ) A 60; w0 20* pi; phi 0; a 6; 0 : 0.001:1; x A*sin( w0* + phi).*exp( a* ); plo(, x) 43
Exp. Damped Sine Signal Plo 44 Audio & DSP Lab.
Sep, Impulse and Ramp Funcion Generae Sep Funcion: u [ zeros(1,50), ones(1,50)]; Generae Impulse Funcion: dela [ zeros(1,49),1, zeros(1,49)]; Generae Ramp Funcion: ramp 0 : 0.1:10; 45
User Defined Funcion Generae.m file o define funcion: funcion g rec( x) g zeros( size( x)); se find( abs( x) < 0.5); g( se) ones( size( se)); 46
P1.46 E + π / ω π / ω x 2 ( ) d x( ) 2 0, [ 1+ cos( ω ) ], π / ω ohers π / 1 ω π ω + π ω 47
x( ) P1.47 5, 1, + 5, 0, 4 4 5 ohers 5 4 4 E + 5 5 x 2 ( ) d 48
( a) ( b) ( c) P1.52 x( ) y( x( x( 1) 1) y( ) + 1) y( 2) 49
P1.53 g() x() 50 Audio & DSP Lab.
P1.55 g() x() 51 Audio & DSP Lab.
P1.56 (1) x[2n] (2) x[3n 1] (3) y[1 n] (4) y[2 2n] (5) x[ n 2] + y[ n + 2] 52
P1.62 (a) y() (b) y() δ() 4 4 2 + + 2 2 2 53
P1.63 54 Audio & DSP Lab.
P1.75 ( ) 55
P1.76 ( ) 56
P1.77 (a) y[n] x[n] δ[n] (b) x[n] y[n] 57
P1.79 58
P1.80 59 Audio & DSP Lab.
P1.81 H 60
P1.82 A? 1 x ( ) + ( + / τ / τ e u( ) e u( ) ) 61
P1.83 (a) (b) 62
P1.84 : (a) y( ) A cos 0 ( ω + φ) x( ) 0 (b) 63
P1.89 x[n] y[n] 64
P1.92 65 Audio & DSP Lab.