http:elec34.org Discrete Systems & Z-Transforms 4 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: eek Date Lecture Title -Mar Introduction 3-Mar Systems Overview 9-Mar Signals as Vectors & Systems as Maps -Mar [Signals] 3 6-Mar Sampling & Data Acquisition & Antialiasing Filters 7-Mar [Sampling] 4 3-Mar System Analysis & Convolution 4-Mar [Convolution & FT] 5 6 7 8 9 3 3-Mar Discrete Systems & Z-Transforms 3-Mar [Z-Transforms] 3-Apr Frequency Response & Filter Analysis 4-Apr [Filters] -Apr Digital Filters -Apr [Digital Filters] 7-Apr Introduction to Digital Control 8-Apr [Feedback] 4-May Digital Control Design 5-May [Digitial Control] -May Stability of Digital Systems -May [Stability] 8-May State-Space 9-May Controllability & Observability 5-May PID Control & System Identification 6-May Digitial Control System Hardware 3-May Applications in Industry & Information Theory & Communications -Jun Summary and Course Review ELEC 34: Systems 3 March 5 -
Lecture Overview Course So Far: ODE Transfer functions Convolution L: Laplace (s) Cascade of LCC ODE Z-Transform Lecture(s): Convolution F: Fourier Series (Periodic functions) L F: (ξ = σ + iτ) (R C) C: Poles & Zeros DFFT Z-Transform ELEC 34: Systems 3 March 5-3 Graphical Understanding of Convolution For c(τ)= :. Keep the function f (τ) fixed. Flip (invert) 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(t) for all values of t. ELEC 34: Systems 3 March 5-4
Graphical Understanding of Convolution (Ex) ELEC 34: Systems 3 March 5-5 Complex umbers and Phasors Positive Frequency component Rsin( ) Y R Rcos( ) Re j X j Re ( Rcos, Rsin ) Rcos jrsin R(cos j sin ) ELEC 34: Systems 3 March 5-6 3
Complex umbers and Phasors egative frequency component Rsin( ) Y R cos( ) R Re j X j Re ( Rcos( ), Rsin( )) Rcos( ) jrsin( ) R(cos j sin ) ELEC 34: Systems 3 March 5-7 Positive and egative Frequencies Frequency is the derivative of phase more nuanced than : = repetition rate τ Hence both positive and negative frequencies are possible. Compare velocity vs speed frequency vs repetition rate ELEC 34: Systems 3 March 5-8 4
egative Frequency Q: hat is negative frequency? A: A mathematical convenience Trigonometrical FS periodic signal is made up from sum to of sine and cosines harmonics Complex Fourier Series & the Fourier Transform use exp(jωt) instead of cos(ωt) and sin(ωt) signal is sum from to of exp(jωt) same as sum - to of exp(-jωt) which is more compact (i.e., less chalk!) ELEC 34: Systems 3 March 5-9 Another way to see Aliasing Too! Rotating wheel and peg eed both top and front view to determine rotation Top View Front View ELEC 34: Systems 3 March 5-5
Frequency Response Fourier Series Fourier Transforms ELEC 34: Systems 3 March 5 - Typical Linear Processors Convolution Cross Correlation Auto Correlation Cosine Transform Sine Transform Fourier Transform h(n,k)=h(n-k) h(n,k)=h(n+k) h(n,k)=x(k-n) h(n,k)= h(n,k)= h(n,k)= cos nk sin nk exp j nk ELEC 34: Systems 3 March 5-6
Transform Analysis Signal measured (or known) as a function of an independent variable e.g., time: y = f(t) However, this independent variable may not be the most appropriateinformative e.g., frequency: Y = f(w) Therefore, need to transform from one domain to the other e.g., time frequency As used by the human ear (and eye) Signal processing uses Fourier, Laplace, & z transforms etc ELEC 34: Systems 3 March 5-3 Sinusoids and Linear Systems x(t) or x(n) h(t) or h(n) y(t) or y(n) If or x( t) Acos( t ) x( n) Acos( nt ) then in steady state y( t) AC( )cos( t ( )) y( n) AC( T)cos( nt ( T)) ELEC 34: Systems 3 March 5-4 7
Sinusoids and Linear Systems The pair of numbers C(ω ) and q(ω ) are the complex gain of the system at the frequency ω. They are respectively, the magnitude response and the phase response at the frequency ω. y( t) AC( )cos( t ( )) y( n) AC( T)cos( nt ( T)) ELEC 34: Systems 3 March 5-5 hy Use Sinusoids? hy probe system with sinusoids? Sinusoids are eigenfunctions of linear systems??? hat the hell does that mean? Sinusoid in implies sinusoid out Only need to know phase and magnitude (two parameters) to fully describe output rather than whole waveform sine + sine = sine derivative of sine = sine (phase shifted - cos) integral of sine = sine (-cos) Sinusoids maintain orthogonality after sampling (not true of most orthogonal sets) ELEC 34: Systems 3 March 5-6 8
Frequency Response Fourier Series Fourier Transforms ELEC 34: Systems 3 March 5-7 Fourier Series Deal with continuous-time periodic signals. Discrete frequency spectra. A Periodic Signal f(t) T T 3T t Source: URI ELE436 ELEC 34: Systems 3 March 5-8 9
Two Forms for Fourier Series Sinusoidal Form Complex Form: a f ( t) a T n T ) T a f ( t dt n nt cos T n nt bn sin T jn t T jnt ( t) c n e cn f t e dt n T ( ) T f a b T T n T T f ( t)cos n tdt T n T f ( t)sin n tdt Source: URI ELE436 ELEC 34: Systems 3 March 5-9 Fourier Series Any finite power, periodic, signal x(t) period T can be represented as () summation of sine and cosine waves Called: Trigonometrical Fourier Series A x t A n t B n t ( ) ncos( ) nsin( ) n Fundamental frequency ω =T rads or T Hz DC (average) value A ELEC 34: Systems 3 March 5 -
Amplitude y = f(t) - - 3 4 5 6 time (t).5.5 3 4 5 6 7 8 frequency (f) Frequency representation (spectrum) shows signal contains: Hz and 5Hz components (sinewaves) of equal amplitude ELEC 34: Systems 3 March 5 - Fourier Series Coefficients An & Bn calculated from the signal, x(t) called: Fourier coefficients A n T T T x( t)cos( nw t) dt n,,, B n T T T x( t)sin( nw t) dt n,,3, ote: Limits of integration can vary, provided they cover one period ELEC 34: Systems 3 March 5 -
harmonic harmonic.5.5 -.5 - signal (black), approximation (red) and harmonic (green) -.5...3.4.5.6.7.8.9 time (s).5.5 -.5 - -.5...3.4.5.6.7.8.9 time (s) ELEC 34: Systems 3 March 5-3.5.5 -.5 - signal (black), approximation (red) and harmonic (green) -.5...3.4.5.6.7.8.9 time (s).5.5 -.5 - -.5...3.4.5.6.7.8.9 time (s) ELEC 34: Systems 3 March 5-4
harmonic harmonic.5.5 -.5 - signal (black), approximation (red) and harmonic (green) -.5...3.4.5.6.7.8.9 time (s).5.5 -.5 - -.5...3.4.5.6.7.8.9 time (s) ELEC 34: Systems 3 March 5-5.5.5 -.5 - signal (black), approximation (red) and harmonic (green) -.5...3.4.5.6.7.8.9 time (s).5.5 -.5 - -.5...3.4.5.6.7.8.9 time (s) ELEC 34: Systems 3 March 5-6 3
Example: Square wave, x( t), x( t ). A B n t ; t ; x( t)cos( nt) dt sin An n n nt sinnt x( t)sin( nt) dt cos( nt) dt n sin( nt) dt cos( nt) dt sin( nt) dt nt cosnt cosn cosn cos Bn n Bn ( cos( n )) n periodic! i.e., x(t + ) = x(t) n n Sin terms only o cos terms as sin(n) = n x(t) has odd symmetry n n cos(n) = n n ELEC 34: Systems 3 March 5-8 Example: Square wave Therefore, Trigonomet ric Fourier series ( ) x t ( cos( n ))sin( nt) n n Expanding the terms gives, x(t) 4 sin( t) 4 sin(3t ) 3 4 sin(5t ) 5 etc (fundamental) (second harmonic) (third harmonic) (fourth harmonic) (fifth harmonic) is, Only odd harmonics; In proportion,3,5,7, Higher harmonics contribute less; Therefore, converges ELEC 34: Systems 3 March 5-9 4
How to Deal with Aperiodic Signal? A Periodic Signal f(t) T t If T, what happens? Source: URI ELE436 ELEC 34: Systems 3 March 5-3 Fourier Integral jnt ft ( t) cne n n T T T f ( ) e T c jn T T jnt n ft ( t) e T d e jnt T dt T n n T jn jnt ft ( ) e d e T T T jn jnt ft ( ) e d e f ( e j T ) de jt d Let T T d Source: URI ELE436 ELEC 34: Systems 3 March 5-3 5
Fourier Integral f ( t) f ( ) e j de jt d F( j ) f ( t) e jt F(j) j t f ( t) F( j ) e d Synthesis dt Analysis Source: URI ELE436 ELEC 34: Systems 3 March 5-3 Fourier Series vs. Fourier Integral Fourier Series: Fourier Integral: n f ( t) c n e c T jn t T jnt n ft ( t) e T f ( t) F( j) e dt j t d Period Function Discrete Spectra on-period Function F( j ) f ( t) e jt dt Continuous Spectra Source: URI ELE436 ELEC 34: Systems 3 March 5-33 6
Complex Fourier Series (CFS) Also called Exponential Fourier series As it uses Euler s relation Aexp( jw t) Acos( w t) jasin( w t) which implies, exp( jnwt) exp( jnwt) cos( nwt) exp( jnwt) exp( jnwt) sin( nwt) j FS as a Complex phasor summation ( t) X n exp( jnw n x t) here X n are the CFS coefficients ELEC 34: Systems 3 March 5-34 Complex Fourier Coefficients Again, Xn calculated from x(t) X n T T T x( t)exp( jnw t Only one set of coefficients, Xn but, generally they are complex ) dt Remember: fundamental ω = T! ELEC 34: Systems 3 March 5-35 7
Relationships There is a simple relationship between trigonometrical and complex Fourier coefficients, X X n A A A n n jb jb n n,, n ; n. Constrained to be symmetrical, i.e., complex conjugate * n X n Therefore, can calculate simplest form and convert ELEC 34: Systems 3 March 5-36 X Example: Complex FS Consider the pulse train signal Has complex Fourier series: A, x( t), x( t T ). t ; t T; A T ote: by X n T T T x( t)exp A jn exp jnt jn tdt Aexp jn t T jn exp dt ote: n is the ind. variable ELEC 34: Systems 3 March 5-37 8
Example: Complex FS X n w exp A sin( nw ) T nw T ote: letting cos cos n hich using Euler s identity reduces to: A sa( nw ) T j exp j j sin cos j sin j sin cos j sin j sin ote: cos(-) = cos(): even sin(-) = -sin(): odd ELEC 34: Systems 3 March 5-38 Frequency Response Fourier Series Fourier Transforms ELEC 34: Systems 3 March 5-54 9
Fourier Transform A Fourier Transform is an integral transform that re-expresses a function in terms of different sine waves of varying amplitudes, wavelengths, and phases. -D Example: hen you let these three waves interfere with each other you get your original wave function! Source: Tufts Uni Sykes Group ELEC 34: Systems 3 March 5-55 Fourier Series hat we have produced is a processor to calculate one coefficient of the complex Fourier Series Fourier Series Coefficients = Heterodyne and average over observation interval T C k T T h t e j kt ( ) T dt ELEC 34: Systems 3 March 5-56
Aperiodic Periodic Fourier Transform If we change the limits of integration to the entire real line, remove the division by T, and make the frequency variable continuous, we get the Fourier Transform j t C ( ) h( t ) e dt ELEC 34: Systems 3 March 5-57 Fourier Transform (is not the Fourier Series per se) Continuous Time Fourier Series Continuous Fourier Transform Discrete Time Discrete Fourier Transform Fourier Transform Source: URI ELE436 ELEC 34: Systems 3 March 5-58
Fourier Transform Fourier series Only applicable to periodic signals Real world signals are rarely periodic Develop Fourier transform by Examining a periodic signal Extending the period to infinity ELEC 34: Systems 3 March 5-59 Fourier Transform Problem: as T, X n i.e., Fourier coefficients vanish! Solution: re-define coefficients X n = T X n As T (harmonic frequency) nω ω (continuous freq.) (discrete spectrum) X n X(ω) (continuous spect.) ω (fundamental freq.) reduces dω (differential) Summation becomes integration ELEC 34: Systems 3 March 5-6
Fourier Transform Pair Inverse Fourier Transform: j f t F j e t ( ) ( ) d Synthesis Fourier Transform: F( j ) f ( t) e jt dt Analysis Source: URI ELE436 ELEC 34: Systems 3 March 5-63 Continuous Spectra F( j ) f ( t) e jt dt F( j) F ( j) jf ( j) R F( j) e Magnitude I j( ) Phase F I (j) () F R (j) Source: URI ELE436 ELEC 34: Systems 3 March 5-64 3
X(w) x(t) X(w) x(t) Pulse width = Time limited.8.6.4. - -8-6 -4-4 6 8 time (t) Infinite bandwidth.5 rect(t) sinc(w) -.5 - -5 5 Angular frequency (w) ELEC 34: Systems 3 March 5-66.8.6.4. Pulse width = - -8-6 -4-4 6 8 time (t) Parseval s Theorem.5.5 rect(t) sinc(w) -.5 - -5 5 Angular frequency (w) ELEC 34: Systems 3 March 5-67 4
X(w) x(t) X(w) x(t).8.6.4. Pulse width = 4 - -8-6 -4-4 6 8 time (t) 4 3 rect(t4) 4 sinc(w) - - -5 5 Angular frequency (w) ELEC 34: Systems 3 March 5-68.8.6.4. Pulse width = 8 - -8-6 -4-4 6 8 time (t) 8 6 4 rect(t8) 8 sinc(4w) - - -5 5 Angular frequency (w) ELEC 34: Systems 3 March 5-69 5
X(w) x(t).5 Symmetry: F{sinc(t)} = rect(-w) sinc(t) Infinite time -.5-4 -3 - - 3 4 time (t) Finite bandwidth rect(-w) -5-4 -3 - - 3 4 5 Angular frequency (w) Ideal Lowpass 3 March 5 filter - ELEC 34: Systems 7 Properties of Fourier Transform Linearity F {a x(t) + b y(t)} = a X(ω) + b Y(ω) Time and frequency scaling F {x(at)} = a X(ωa) broader in time narrower in frequency and vice versa Symmetry (duality) x(-ω) = X(t) exp(-jωt)dt i.e., Fourier transform pairs Time limited signal limited has infinite bandwidth; Signal of finite bandwidth has infinite time support ELEC 34: Systems 3 March 5-7 6
X(w) x(t) Properties of Fourier Transform if x(t) is real x(t) is real and even x(t) is real and odd Then X(-ω) = X(ω)* {X(ω)} is even {X(ω)} is odd X(ω) is even X(ω) is odd X(ω) is real and even X(ω) is imaginary and odd ELEC 34: Systems 3 March 5-7 Fourier Transforms X t exp t.5 -.5 x(t) = cos(ω t) (real & even) - -5 - -5 5 5 t 3 X(w) = [(ω-ω ) + (ω+ω )] (real and even) -ω ω ω ote: cos(ωt) has energy! But is dual of (w ω) ELEC 34: Systems 3 March 5-73 7
{X(w)} x(t) Fourier Transforms.5 -.5 - -5 - -5 5 5 t x(t) = sin(ω t) (real and odd) 4 - -4 -ω ω 3 4 X(w) = j[(ω+ω ) - (ω-ω )] (imaginary & odd) ote: sin & cos have same Mag spectrum, Phase is only difference ELEC 34: Systems 3 March 5-74 Properties of Fourier Transform Time Shift F {x(t - )} = exp(-j w)x(w) time shift phase shift Convolution and multiplication F {x(t) y(t)} = X(ω) Y(ω) i.e., implement convolution in Fourier domain F {x(t) y(t)} = (X(ω) Y(ω)) i.e., Fourier interpretation of multiplication (e.g., frequency modulation) ELEC 34: Systems 3 March 5-75 8
phase (degrees) exp(-j ) Magnitude and Phase of exp(-j w)..8.6 as cos + sin =.4. -5 - -5 5 5 Frequency (rads) - - -5 - -5 5 5 Frequency (rads) ELEC 34: Systems 3 March 5-76 modifies phase only x(t) = rect(t) X(w) = sinc(w) ZOH.5 F.5 h(t) = rect(t) -.5 H(w) = sinc(w) ZOH.5 F.5 x(t)*h(t) = tri(t) -.5 X(w)H(w) = sinc (w) FOH F.5 - - - ELEC 34: Systems 3 March 5-77 9
More properties of the FT Differentiation in time Integration in time d F x t j X dt n d F x t j X dt n Differentiation ω (ote: HPF & DC x zero) t F x t dt X X j Integration ω + DC offset (LPF & opposite of differentiation) ELEC 34: Systems 3 March 5-78 More Fourier Transforms f(t) =(t - nt) = (t) F(w) = (t)(ω - nt) F t f(t) = (t - nt) F t ω Impulse train, comb or Shah function ω F(w) = (t) (ω - nt) ELEC 34: Systems 3 March 5-79 3
More Fourier Transforms Limit of previous as t and t respectively ote: f(t) = has energy! But is dual of (t) f(t) = (t) F(ω) = F f(t) = t F ω F(ω) = (ω) t ote: u(t) also has energy! But F{u(t)} = F{(t)} i.e., apply integration property ω ELEC 34: Systems 3 March 5-8 Pop Quiz : Questions If F{x(t)} = X(w) F{x(t)} =? F{x(t4)} =? F{(t)} =? F{} =? ELEC 34: Systems 3 March 5-8 3
Pop Quiz: Answers! o orries LectopiaLand! If F{x(t)} = X(ω) F{x(t)} = X(ω) narrower in t broader in freq F{x(t4)} = 4X(4ω) broader in t narrower in freq (but increased amplitude) F{(t)} = i.e. flat spectrum (all frequencies equally) F{} = (ω) i.e. impulse at DC only ELEC 34: Systems 3 March 5-8 Interpretation of Fourier Transform Represents (usually finite energy) signals as sum of cosine waves at all possible frequencies X(ω) dω is amplitude of cosine wave i.e., in frequency band ω to ω + dω X(ω) is phase shift of cosine wave Also represents finite power, periodic signals Using (ω) Distribution with frequency of both magnitude & phase called a Frequency spectrum (continuous) ELEC 34: Systems 3 March 5-83 3
Magnitude Fourier Image Examples Lena Bridge ELEC 34: Systems 3 March 5-84 Fourier Magnitude and Phase *log(abs(fft(lena))) angle(fft(lena)).5 3 4 5 6 7 8 9 Frequency f random range() Bridge spectra look similar ELEC 34: Systems 3 March 5-85 33
Magnitude and Phase Only ifft(abs(fft(lena)) + angle()) ifft(abs(fft(bridge)) + angle(fft(lena))) Lena magnitude only Lena phase + bridge magnitude ote: titles are illustrative only and are not the actual Matlab commands used! ELEC 34: Systems 3 March 5-86 FFT Fourier Transform (not just yet we ll come back to this) ELEC 34: Systems 3 March 5-98 34
Fourier Analysis of Discrete Time Signals For a discrete time sequence we define two classes of Fourier Transforms: the DTFT (Discrete Time FT) for sequences having infinite duration, the DFT (Discrete FT) for sequences having finite duration. ELEC 34: Systems 3 March 5-99 The Discrete Time Fourier Transform (DTFT) Given a sequence x(n) having infinite duration, we define the DTFT as follows: X ( ) DTFT x( n) x( n) e n j n j n x( n) IDTFTX ( ) X ( ) e d x( n) X ( ).. discrete time.. n continuous frequency ELEC 34: Systems 3 March 5-35
Observations: The DTFT X () is periodic with period ; The frequency is the digital frequency and therefore it is limited to the interval Recall that the digital frequency is a normalized frequency relative to F the sampling frequency, defined as F s one period of X ( ) F s F s F s Fs F ELEC 34: Systems 3 March 5 - Example x[n] DTFT n since X ( ) e n j n e e e j j j ( ) sin sin ELEC 34: Systems 3 March 5-36
Fast Fourier Transform Algorithms Consider DTFT Basic idea is to split the sum into subsequences of length and continue all the way down until you have subsequences of length Log(8) ELEC 34: Systems 3 March 5-3 Radix- FFT Algorithms - Two point FFT e assume =^m This is called Radix- FFT Algorithms Let s take a simple example where only two points are given n=, n=; = Butterfly FFT y y Advantage: Less computationally intensive: O(*log()) Source: http:www.cmlab.csie.ntu.edu.twcmldsptrainingcodingtransformfft.html y ELEC 34: Systems 3 March 5-4 37
38 First break x[n] into even and odd Let n=m for even and n=m+ for odd Even and odd parts are both DFT of a point sequence Break up the size subsequent in half by letting mm The first subsequence here is the term x[], x[4], The second subsequent is x[], x[6], ) sin( ) cos( ]) [ ( ] [ j m m m mk mk m mk k m mk j e m x m x General FFT Algorithm 3 March 5 - ELEC 34: Systems 5 Example ) sin( ) cos( ]) [ ( ] [ ] [ j m m m mk mk m mk k m mk j e m x m x k X Let s take a simple example where only two points are given n=, n=; = [] [] [] [] []) ( [] ] [ [] [] []) ( [] ] [.... x x x x x x k X x x x x k X m m m m Same result 3 March 5 - ELEC 34: Systems 6
FFT Algorithms - Four point FFT First find even and odd parts and then combine them: The general form: ELEC 34: Systems 3 March 5-7 FFT Algorithms - 8 point FFT Applet: http:www.falstad.comfourierdirections.html http:www.engineeringproductivitytools.comstufftpt7.htm ELEC 34: Systems 3 March 5-8 39
Poles and Zeros ELEC 34: Systems 3 March 5-9 Poles and Zeros Source: Boyd, EE,5- ELEC 34: Systems 3 March 5-4
Poles and Zeros Source: Boyd, EE,5-3 ELEC 34: Systems 3 March 5 - Pole Zero Plot Source: Boyd, EE,5-4 ELEC 34: Systems 3 March 5-4
Partial Fraction Expansion Source: Boyd, EE,5-5 ELEC 34: Systems 3 March 5-3 Partial Fraction Expansion Example Source: Boyd, EE,5-6 ELEC 34: Systems 3 March 5-4 4
How to Handle the Digitization? (z-transforms) ELEC 34: Systems 3 March 5-5 The z-transform The discrete equivalent is the z-transform : and Z f k = f(k)z k = F z k= Z f k = z F z x(k) F(z) Convenient! y(k) This is not an approximation, but approximations are easier to derive ELEC 34: Systems 3 March 5-6 43
The z-transform It is defined by: Or in the Laplace domain: z = e st Thus: or I.E., It s a discrete version of the Laplace: f kt = e akt z Z f k = z e at ELEC 34: Systems 3 March 5-7 The z-transform In practice, you ll use look-up tables or computer tools (ie. Matlab) to find the z-transform of your functions F(s) F(kt) F(z) s z z kt Tz s z e akt z s + a z e at kte akt zte at s + a z e at sin (akt) z sin at s + a z cos at z + ELEC 34: Systems 3 March 5-8 44
L(ZOH)=??? : hat is it? ikipedia Lathi Franklin, Powell, orkman Franklin, Powell, Emani-aeini Dorf & Bishop Oxford Discrete Systems: (Mark Cannon) MIT 6. (Russ Tedrake) Matlab Proof! ELEC 34: Systems 3 March 5-9 Zero-order-hold (ZOH) x(t) x(kt) h(t) Zero-order Hold Sampler Assume that the signal x(t) is zero for t<, then the output h(t) is related to x(t) as follows: ELEC 34: Systems 3 March 5-45
Transfer function of Zero-order-hold (ZOH) Recall the Laplace Transforms (L) of: Thus the L of h(t) becomes: ELEC 34: Systems 3 March 5 - Transfer function of Zero-order-hold (ZOH) Continuing the L of h(t) Thus, giving the transfer function as: Z ELEC 34: Systems 3 March 5-46