http://elec34.org Sampling and CONVOLUTION 24 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 2-Mar Introduction 3-Mar Systems Overview 2 3 9-Mar Signals as Vectors & Systems as Maps -Mar [Signals] 6-Mar Sampling & Data Acquisition & Antialiasing Filters 7-Mar [Sampling] 4 23-Mar Sampling & Convolution 5 6 7 8 9 2 3 24-Mar [Convolution & FT] 3-Mar Frequency Response & Filter Analysis 3-Mar [Filters] 3-Apr Discrete Systems & Z-Transforms 4-Apr [Z-Transforms] 2-Apr Introduction to Digital Control 2-Apr [Feedback] 27-Apr Digital Filters 28-Apr [Digital Filters] 4-May Digital Control Design 5-May [Digitial Control] -May Stability of Digital Systems 2-May [Stability] 8-May State-Space 9-May Controllability & Observability 25-May PID Control & System Identification 26-May Digitial Control System Hardware 3-May Applications in Industry & Information Theory & Communications 2-Jun Summary and Course Review ELEC 34: Systems 23 March 25-2
signal Sampling Theorem The Nyquist criterion states: To prevent aliasing, a bandlimited signal of bandwidth w B rad/s must be sampled at a rate greater than 2w B rad/s w s > 2w B Note: this is a > sign not a Also note: Most real world signals require band-limiting with a lowpass (anti-aliasing) filter ELEC 34: Systems 23 March 25-3 Sampling < Nyquist Aliasing.5.5 -.5 - True signal Aliased (under sampled) signal -.5 5 5 time ELEC 34: Systems 23 March 25-4 2
Reconstruction ELEC 34: Systems 23 March 25-5 Reconstruction Zero-Order Hold [ZOH] ELEC 34: Systems 23 March 25-6 3
X(w) x( Reconstruction Whittaker Shannon interpolation formula ELEC 34: Systems 23 March 25-7.5 Symmetry: F{sinc(t/2)} = 2 rect(-w) sinc(t/2) Infinite time -.25-4 -3-2 - 2 3 4 time ( Finite bandwidth 2 2 rect(-w) -5-4 -3-2 - 2 3 4 5 Angular frequency (w) Ideal Lowpass 23 March 25 filter - ELEC 34: Systems 8 4
X(w) x( X(w) x( Pulse width = Time limited.8.6.4.2 - -8-6 -4-2 2 4 6 8 time ( Infinite bandwidth.5 rect( sinc(w/2) -.5-2 -5 5 2 Angular frequency (w) ELEC 34: Systems 23 March 25-9.8.6.4.2 Pulse width = 2 - -8-6 -4-2 2 4 6 8 time ( Parseval s Theorem 2.5.5 rect(t/2) 2 sinc(w/) -.5 - -5 5 Angular frequency (w) ELEC 34: Systems 23 March 25-5
X(w) x( X(w) x(.8.6.4.2 Pulse width = 4 - -8-6 -4-2 2 4 6 8 time ( 4 3 2 rect(t/4) 4 sinc(2w/) - - -5 5 Angular frequency (w) ELEC 34: Systems 23 March 25 -.8.6.4.2 Pulse width = 8 - -8-6 -4-2 2 4 6 8 time ( 8 6 4 2 rect(t/8) 8 sinc(4w/) -2 - -5 5 Angular frequency (w) ELEC 34: Systems 23 March 25-2 6
Reconstruction Whittaker Shannon interpolation formula ELEC 34: Systems 23 March 25-3 Original Signal After Anti-aliasing LPF After Sample & Hold After Reconstruction LPF After D/A After A/D Complete practical DSP system signals DSP ELEC 34: Systems 23 March 25-4 7
Convolution ELEC 34: Systems 23 March 25-5 Convolution Definition The convolution of two functions f ( and f 2 ( is defined as: f ( f( ) t ) d f( * Source: URI ELE436 ELEC 34: Systems 23 March 25-6 8
Convolution: Concepts My goal is to give you a feel for the Convolution in Systems For the mechanics of Convolution: Many good Convolution reviews online EG: Khan Academy https://www.khanacademy.org/math/differential-equations/laplace-transform/convolution-integral/v/the-convolution-and-the-laplace-transform & More http://graphics.stanford.edu/courses/cs78/applets/convolution.html ELEC 34: Systems 23 March 25-7 Convolution & Properties Properties: Commutative: Distributive: Associative: Shift: if f (*f 2 (=c(, then f (t-t)*f 2 (= f (*f 2 (t-t)=c(t-t) Identity (Convolution with an Impulse): Total Width: Based on Lathi, SPLS, Sec 2.4- ELEC 34: Systems 23 March 25-8 9
Convolution & Properties [II] Convolution systems are linear: Convolution systems are causal: the output y( at time t depends only on past inputs Convolution systems are time-invariant (if we shift the signal, the output similarly shifts) ELEC 34: Systems 23 March 25-9 Convolution & Properties [III] Composition of convolution systems corresponds to: multiplication of transfer functions convolution of impulse responses Thus: We can manipulate block diagrams with transfer functions as if they were simple gains convolution systems commute with each other ELEC 34: Systems 23 March 25-2
Properties of Convolution f( * * f( f ( * f 2( f( ) t ) d f ( ) t ) d t t f( t ) f2[ t ( t )] d( t ) f t ) f ( ) d ( 2 ( 2 * f( f t ) f ( ) d Source: URI ELE436 ELEC 34: Systems 23 March 25-2 Properties of Convolution f( f( * * f( Impulse Response LTI System h( f(*h( h( Impulse Response LTI System f( h(*f( Source: URI ELE436 ELEC 34: Systems 23 March 25-22
Properties of Convolution [ 3 f( * ]* f3( f( *[ * f ( ] h ( h 2 ( h 3 ( h 2 ( h 3 ( h ( The two systems are identical! Source: URI ELE436 ELEC 34: Systems 23 March 25-23 Properties of Convolution f ( * ( f ( f( ( f( f ( * ( f ( ) ( t ) d f ( f ( t ) ( ) d Source: URI ELE436 ELEC 34: Systems 23 March 25-24 2
Properties of Convolution f ( * ( f ( f( ( f( f ( * ( t T) f ( t T) f ( * ( t T) f ( ) ( t T ) d f ( t T ) ( ) d f ( t T) Source: URI ELE436 ELEC 34: Systems 23 March 25-25 Properties of Convolution f ( * ( t T) f ( t T) f( (tt) T f(t T) f ( t T f ( t Source: URI ELE436 ELEC 34: Systems 23 March 25-26 3
Properties of Convolution F f( * F ( j) F2 ( j) jt F[ f t f t f f t d ( )* 2( )] e dt ( ) 2( ) f ( j t ) t ) e dt d Time Domain convolution f ) F ( 2 ( j) e j d j F2 ( j) f( ) e d F ( j) F2 ( j) Frequency Domain multiplication Source: URI ELE436 ELEC 34: Systems 23 March 25-27 Properties of Convolution F f( * F ( j) F2 ( j) F i (j) H(j) F o (j) p p An Ideal Low-Pass Filter Source: URI ELE436 ELEC 34: Systems 23 March 25-28 4
Properties of Convolution F f( * F ( j) F2 ( j) F i (j) H(j) F o (j) p p An Ideal High-Pass Filter Source: URI ELE436 ELEC 34: Systems 23 March 25-29 Convolution & Systems Convolution system with input u (u( =, t < ) and output y: abbreviated: in the frequency domain: ELEC 34: Systems 23 March 25-3 5
Systems Interpretation Source: Lecture Notes for EE263, Stephen Boyd, Stanford 22., Slide: 3-6 ELEC 34: Systems 23 March 25-3 Systems Interpretation Source: Lecture Notes for EE263, Stephen Boyd, Stanford 22., Slide: 3-7 ELEC 34: Systems 23 March 25-32 6
Systems Interpretation Source: Lecture Notes for EE263, Stephen Boyd, Stanford 22., Slide: 3-8 ELEC 34: Systems 23 March 25-33 Systems Interpretation Source: Lecture Notes for EE263, Stephen Boyd, Stanford 22., Slide: 3-9 ELEC 34: Systems 23 March 25-34 7
Systems Interpretation Source: Lecture Notes for EE263, Stephen Boyd, Stanford 22., Slide: 3- ELEC 34: Systems 23 March 25-35 Convolution & Feedback In the time domain: In the frequency domain: Y=G(U-Y) Y(s) = H(s)U(s) ELEC 34: Systems 23 March 25-36 8
Graphical Understanding of Convolution For c(τ)= :. Keep the function f (τ) fixed 2. Flip (inver the function g(τ) about the vertical axis (τ=) = this is g(-τ) 3. Shift this frame (g(-τ)) along τ (horizontal axis) by t. = this is g(t -τ) For c(t ): 4. c(t ) = the area under the product of f (τ) and g(t -τ) 5. Repeat this procedure, shifting the frame by different values (positive and negative) to obtain c( for all values of t. ELEC 34: Systems 23 March 25-37 Graphical Understanding of Convolution (Ex) ELEC 34: Systems 23 March 25-38 9
Another View e.g. convolution x(k) h(n,k) y(n,k) x(n) = 2 3 4 5 h(n) = 3 2 2 3 4 5 2 3 2 3 4 5 2 3 2 3 4 5 2 3 3 2 6 4 9 y(n) 3 8 4 Sum over all k h(n-k) Notice the gain ELEC 34: Systems 23 March 25-39 Matrix Formulation of Convolution y Hx 3 2 3 8 2 3 4 2 3 2 2 3 26 2 3 4 2 5 Toeplitz Matrix 2. 3 4 3 5 2 3 ELEC 34: Systems 23 March 25-4 2