Frequency Response & Filter Analysis. Week Date Lecture Title. 4-Mar Introduction & Systems Overview 6-Mar [Linear Dynamical Systems] 2

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1 Frequency Response & Filter Analysis 4 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 4-Mar Introduction & Systems Overview 6-Mar [Linear Dynamical Systems] -Mar Signals as Vectors & Systems as Maps 3-Mar [Signals] 3 8-Mar Sampling & Data Acquisition & Antialiasing Filters -Mar [Sampling] 4 5-Mar System Analysis & Convolution 7-Mar [Convolution & FT] 5 -Apr Frequency Response & Filter Analysis Apr [Filters] 8-Apr Discrete Systems & Z-Transforms -Apr [Z-Transforms] 5-Apr Introduction to Control 7-Apr [Feedback] 9-Apr Digital Filters -May [Digital Filters] 6-May Introduction to Digital Control 8-May [Digitial Control] 3-May Stability of Digital Systems 5-May [Stability] -May State-Space -May Controllability & Observability 7-May PID Control & System Identification 9-May Digitial Control System Hardware 3-Jun Applications in Industry & Information Theory & Communications 5-Jun Summary and Course Review ELEC 34: Systems April 4 -

2 Frequency Response Fourier Series Fourier Transforms ELEC 34: Systems April 4-3 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 N nk sin N nk exp j nk N ELEC 34: Systems April 4-4

3 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 appropriate/informative 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 April 4-5 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 April 4-6 3

4 Sinusoids and Linear Systems The pair of numbers C(w) and q(w) are the complex gain of the system at the frequency w. They are respectively, the magnitude response and the phase response at the frequency w. y( t) AC( )cos( t ( )) y( n) AC( T)cos( nt ( T)) ELEC 34: Systems April 4-7 Why Use Sinusoids? Why probe system with sinusoids? Sinusoids are eigenfunctions of linear systems??? What 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 April 4-8 4

5 Frequency Response Fourier Series Fourier Transforms ELEC 34: Systems April 4-9 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 April 4-5

6 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 April 4 - 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) An cos( nwt) Bn sin( nw t n ) Fundamental frequency w =/T rad/s or /T Hz DC (average) value A / ELEC 34: Systems April 4-6

7 Amplitude y = f(t) time (t) frequency (f) Frequency representation (spectrum) shows signal contains: Hz and 5Hz components (sinewaves) of equal amplitude ELEC 34: Systems April 4-3 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, Note: Limits of integration can vary, provided they cover one period ELEC 34: Systems April 4-4 7

8 harmonic harmonic signal (black), approximation (red) and harmonic (green) time (s) time (s) ELEC 34: Systems April signal (black), approximation (red) and harmonic (green) time (s) time (s) ELEC 34: Systems April 4-6 8

9 harmonic harmonic signal (black), approximation (red) and harmonic (green) time (s) time (s) ELEC 34: Systems April signal (black), approximation (red) and harmonic (green) time (s) time (s) ELEC 34: Systems April 4-8 9

10 Fourier Series Coefficients Approximation with st, 3rd, 5th, & 7th Harmonics added, note: Ringing on edges due to series truncation Often referred to as Gibb s phenomenon Fourier series converges to original signal if Dirichlet conditions satisfied Closer approximation with more harmonics ELEC 34: Systems April 4-9 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 No cos terms as sin(n) = n x(t) has odd symmetry n n cos(n) = n n ELEC 34: Systems April 4 -

11 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 April 4 - How to Deal with Aperiodic Signal? A Periodic Signal f(t) T t If T, what happens? Source: URI ELE436 ELEC 34: Systems April 4 -

12 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 April 4-3 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 April 4-4

13 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 Non-Period Function F( j ) f ( t) e jt dt Continuous Spectra Source: URI ELE436 ELEC 34: Systems April 4-5 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) Where X n are the CFS coefficients ELEC 34: Systems April 4-6 3

14 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 w = /T! ELEC 34: Systems April 4-7 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 April 4-8 X 4

15 Example: Complex FS Consider the pulse train signal Has complex Fourier series: A, x( t), x( t T ). t ; t T; A T Note: by / X n T T T x( t)exp A jn exp jnt jn tdt Aexp jn t T jn exp dt Note: n is the ind. variable ELEC 34: Systems April 4-9 Example: Complex FS X n w exp A sin( nw ) T nw T Note: letting cos cos n Which 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 Note: cos(-) = cos(): even sin(-) = -sin(): odd ELEC 34: Systems April 4-3 5

16 Dirichlet Conditions For Fourier series to converge, f(t) must be: defined & single valued continuous and have a finite number of finite discontinuities within a periodic interval, and piecewise continuous in periodic interval, as must f (t) be absolutely integrable; i.e., i.e., have finite energy have a finite number of finite discontinuities within a finite interval, and have a finite number of maxima and minima within a finite interval ELEC 34: Systems April 4-35 f ( t) dt Note: Periodic signals have FT, if we use impulse functions, (w) Frequency Response Fourier Series Fourier Transforms ELEC 34: Systems April

17 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: When you let these three waves interfere with each other you get your original wave function! Source: Tufts Uni Sykes Group ELEC 34: Systems April 4-47 Fourier Series What 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 April

18 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 April 4-49 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 April 4-5 8

19 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 April 4-5 Fourier Transform Problem: as T, Xn i.e., Fourier coefficients vanish! Solution: re-define coefficients Xn = T x Xn As T (harmonic frequency) nw w (continuous freq.) (discrete spectrum) Xn X(w) (continuous spect.) w (fundamental freq.) reduces dw (differential) Summation becomes integration ELEC 34: Systems April 4-5 9

20 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 April 4-55 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 April 4-56

21 X(w) x(t) X(w) x(t) Pulse width = Time limited time (t) Infinite bandwidth.5 rect(t) sinc(w/) Angular frequency (w) ELEC 34: Systems April Pulse width = time (t) Parseval s Theorem.5.5 rect(t/) sinc(w/) Angular frequency (w) ELEC 34: Systems April 4-59

22 X(w) x(t) X(w) x(t) Pulse width = time (t) 4 3 rect(t/4) 4 sinc(w/) Angular frequency (w) ELEC 34: Systems April Pulse width = time (t) rect(t/8) 8 sinc(4w/) Angular frequency (w) ELEC 34: Systems April 4-6

23 X(w) x(t).5 Symmetry: F{sinc(t/)} = rect(-w) sinc(t/) Infinite time time (t) Finite bandwidth rect(-w) Angular frequency (w) Ideal Lowpass April 4 filter - ELEC 34: Systems 6 Properties of Fourier Transform Linearity F {a x(t) + b y(t)} = a X(w) + b Y(w) Time and frequency scaling F {x(at)} = /a X(w/a) broader in time narrower in frequency and vice versa Symmetry (duality) x(-w) = X(t) exp(-jwt)dt i.e., Fourier transform pairs Time limited signal limited has infinite bandwidth; Signal of finite bandwidth has infinite time support ELEC 34: Systems April

24 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(-w) = X(w)* {X(w)} is even {X(w)} is odd X(w) is even X(w) is odd X(w) is real and even X(w) is imaginary and odd ELEC 34: Systems April 4-64 Fourier Transforms Note: cos(w t) has energy! But is dual of (w w ) X x t F X x t exp j t x(t) = cos(w t) (real & even) t 3 X(w) = [(w-w ) + (w+w )] (real and even) -w ELEC 34: Systems w April 4-65 w 4

25 {X(w)} x(t) Fourier Transforms Note: sin & cos have same Mag spectrum Phase is only difference t x(t) = sin(w t) (real and odd) w w 3 4 X(w) = j[(w+w ) - (w-w )] (imaginary & odd) ELEC 34: Systems April 4-66 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(w) Y(w) i.e., implement convolution in Fourier domain F {x(t) y(t)} = / (X(w) Y(w)) i.e., Fourier interpretation of multiplication (e.g., frequency modulation) ELEC 34: Systems April

26 phase (degrees) exp(-j ) Magnitude and Phase of exp(-j w)..8.6 as cos + sin = Frequency (rad/s) Frequency (rad/s) ELEC 34: Systems April 4-68 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 ELEC 34: Systems April

27 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 (Note: HPF & DC x zero) t F x t dt X X j Integration /ω + DC offset (LPF & opposite of differentiation) ELEC 34: Systems April 4-7 More Fourier Transforms See Tutorial for proof f(t) =(t - nt) = (t) F(w) = (/t)(w - n/t) F t f(t) = (t - nt) F t w Impulse train, comb or Shah function w F(w) = (/t) (w - n/t) ELEC 34: Systems April 4-7 7

28 More Fourier Transforms Limit of previous as t and t respectively Note: f(t) = has energy! But is dual of (t) f(t) = (t) F(w) = F f(t) = t F w F(w) = (w) t Note: u(t) also has energy! But F{u(t)} = F{(t)} i.e., apply integration property w ELEC 34: Systems April 4-7 Interpretation of Fourier Transform Represents (usually finite energy) signals as sum of cosine waves at all possible frequencies X(w) dw/ is amplitude of cosine wave i.e., in frequency band w to w + dw X(w) is phase shift of cosine wave Also represents finite power, periodic signals Using (w) Distribution with frequency of both magnitude & phase called a Frequency spectrum (continuous) ELEC 34: Systems April

29 Negative Frequency Q: What is negative frequency? A: A mathematical convenience Trigonometrical FS periodic signal is made up from sum to of sine and cosines harmonics Complex FS and the FT use exp(jwt) instead of cos(wt) and sin(wt) signal is sum from to of exp(jwt) same as sum - to of exp(-jwt) which is more compact (i.e., less chalk!) ELEC 34: Systems April 4-74 Negative Frequency imag A wt Ae jwt real Ae jwt A cos( wt) +ve frequency Acos( wt) jasin( wt) Acos( wt) j sin( wt) jsin( wt) -ve frequency ELEC 34: Systems April

30 Magnitude Fourier Image Examples Lena Bridge ELEC 34: Systems April 4-76 Fourier Magnitude and Phase Bridge spectra look similar *log(abs(fft(lena))) angle(fft(lena)) /f.5 random range() ELEC 34: Systems Frequency April

31 Magnitude and Phase Only ifft(abs(fft(lena)) + angle()) ifft(abs(fft(bridge)) + angle(fft(lena))) Lena magnitude only Lena phase + bridge magnitude ELEC 34: Systems Note: titles are illustrative only and are not the actual Matlab commands used! April 4-78 Questions If F{x(t)} = X(w) F{x(t)} =? F{x(t/4)} =? F{(t)} =? F{} =? ELEC 34: Systems April

32 Questions If F{x(t)} = X(w) F{x(t)} = /X(w/) narrower in t broader in freq F{x(t/4)} = 4X(4w) broader in t narrower in freq (but increased amplitude) F{(t)} = i.e. flat spectrum (all frequencies equally) F{} = (w) i.e. impulse at DC only ELEC 34: Systems April 4-8 Frequency How often the signal repeats Can be analyzed through Fourier Transform signal (t) signal(f) Examples: time frequency ELEC 34: Systems 7 April 3-7 3

33 Noise Source: Prof. M. Siegel, CMU Note: this picture illustrates the concepts but it is not quantitatively precise ELEC 34: Systems 7 April 3-8 Noise [] Various Types: Thermal (white): Johnson noise, from thermal energy inherent in mass. Flicker or /f noise: Pink noise More noise at lower frequency Shot noise: Noise from quantum effects as current flows across a semiconductor barrier Avalanche noise: Noise from junction at breakdown (circuit at discharge) ELEC 34: Systems 7 April

34 How to beat the noise Filtering (Narrow-banding): Only look at particular portion of frequency space Multiple measurements Other (modulation, etc.) phase signal noise frequency ELEC 34: Systems 7 April 3 - Noise Uncertainty Uncertainty: All measurement has some approximation A. Statistical uncertainty: quantified by mean & variance B. Systematic uncertainty: non-random error sources Law of Propagation of Uncertainty Combined uncertainty is root squared ELEC 34: Systems 7 April 3-34

35 Treating Uncertainty with Multiple Measurements. Over time: multiple readings of a quantity over time stationary or ergodic system Sometimes called integrating. Over space: single measurement (summed) from multiple sensors each distributed in space 3. Same Measurand: multiple measurements take of the same observable quantity by multiple, related instruments e.g., measure position & velocity simultaneously Basic sensor fusion. ELEC 34: Systems 7 April 3 - Multiple Measurements Example What time was it when this picture was taken? What was the temperature in the room? ELEC 34: Systems 7 April

36 Filters Lowpass Bandpass Highpass Bandstop (Notch) Frequency-shaping filters: LTI systems that change the shape of the spectrum Frequency-selective filters: Systems that pass some frequencies undistorted and attenuate others ELEC 34: Systems 7 April 3-4 Filters Lowpass Specified Values: Gp = minimum passband gain Typically: Highpass Gs = maximum stopband gain Low, not zero (sorry!) For realizable filters, the gain cannot be zero over a finite band (Paley- Wiener condition) Transition Band: transition from the passband to the stopband ωp ωs ELEC 34: Systems 7 April

37 Filter Design & z-transform ELEC 34: Systems April 4-6 Butterworth Filters Butterworth: Smooth in the pass-band The amplitude response H(jω) of an n th order Butterworth low pass filter is given by: The normalized case (ω c =) Recall that: ELEC 34: Systems April

38 Butterworth Filters ELEC 34: Systems April 4-8 Butterworth Filters of Increasing Order: Seeing this Using a Pole-Zero Diagram Increasing the order, increases the number of poles: Odd orders (n=,3,5 ): Have a pole on the Real Axis Angle between poles: Even orders (n=,4,6 ): Have a pole on the off axis ELEC 34: Systems April

39 Butterworth Filters: Pole-Zero Diagram Since H(s) is stable and causal, its poles must lie in the LHP Poles of -H(s) are those in the RHP Poles lie on the unit circle (for a normalized filter) Where: n is the order of the filter ELEC 34: Systems April 4 - Butterworth Filters: 4 th Order Filter Example Plugging in for n=4, k=, 4: We can generalize Butterworth Table This is for 3dB bandwidth at ω c = ELEC 34: Systems April 4-39

40 Butterworth Filters: Scaling Back (from Normalized) Start with Normalized equation & Table Replace ω with in the filter equation For example: for f c =Hz ω c =π rad/sec From the Butterworth table: for n=, a = Thus: ELEC 34: Systems April 4 - Butterworth: Determination of Filter Order Define G x as the gain of a lowpass Butterworth filter at ω= ω x Then: And thus: Or alternatively: & Solving for n gives: PS. See Lathi 4. (p. 453) for an example in MATLAB ELEC 34: Systems April 4-3 4

41 Chebyshev Filters equal-ripple: Because all the ripples in the passband are of equal height If we reduce the ripple, the passband behaviour improves, but it does so at the cost of stopband behaviour ELEC 34: Systems April 4-4 Chebyshev Filters Chebyshev Filters: Provide tighter transition bands (sharper cutoff) than the sameorder Butterworth filter, but this is achieved at the expense of inferior passband behavior (rippling) For the lowpass (LP) case: at higher frequencies (in the stopband), the Chebyshev filter gain is smaller than the comparable Butterworth filter gain by about 6(n - ) db The amplitude response of a normalized Chebyshev lowpass filter is: Where Cn(ω), the nth-order Chebyshev polynomial, is given by: and where C n is given by: ELEC 34: Systems April 4-5 4

42 Normalized Chebyshev Properties It s normalized: The passband is <ω< Amplitude response: has ripples in the passband and is smooth (monotonic) in the stopband Number of ripples: there is a total of n maxima and minima over the passband <ω< ϵ: ripple height The Amplitude at ω=: For Chebyshev filters, the ripple r db takes the place of G p ELEC 34: Systems April 4-6 Determination of Filter Order The gain is given by: Thus, the gain at ω s is: Solving: General Case: ELEC 34: Systems April 4-7 4

43 Chebyshev Pole Zero Diagram Whereas Butterworth poles lie on a semi-circle, The poles of an n th -order normalized Chebyshev filter lie on a semiellipse of the major and minor semiaxes: And the poles are at the locations: ELEC 34: Systems April 4-8 Ex: Chebyshev Pole Zero Diagram for n=3 Procedure:. Draw two semicircles of radii a and b (from the previous slide).. Draw radial lines along the corresponding Butterworth angles (π/n) and locate the n th -order Butterworth poles (shown by crosses) on the two circles. 3. The location of the k th Chebyshev pole is the intersection of the horizontal projection and the vertical projection from the corresponding k th Butterworth poles on the outer and the inner circle, respectively. ELEC 34: Systems April

44 Chebyshev Values / Table ELEC 34: Systems April 4-3 Other Filter Types: Chebyshev Type II = Inverse Chebyshev Filters Chebyshev filters passband has ripples and the stopband is smooth. Instead: this has passband have smooth response and ripples in the stopband. Exhibits maximally flat passband response and equi-ripple stopband Cheby in MATLAB Where: H c is the Chebyshev filter system from before Passband behavior, especially for small ω, is better than Chebyshev Smallest transition band of the 3 filters (Butter, Cheby, Cheby) Less time-delay (or phase loss) than that of the Chebyshev Both needs the same order n to meet a set of specifications. $$$ (or number of elements): Cheby < Inverse Chebyshev < Butterworth (of the same performance [not order]) ELEC 34: Systems April

45 Other Filter Types: Elliptic Filters (or Cauer) Filters Allow ripple in both the passband and the stopband, we can achieve tighter transition band Where: R n is the n th -order Chebyshev rational function determined from a given ripple spec. ϵ controls the ripple Gp = Most efficient (η) the largest ratio of the passband gain to stopband gain or for a given ratio of passband to stopband gain, it requires the smallest transition band in MATLAB: ellipord followed by ellip ELEC 34: Systems April 4-3 In Summary Filter Type Passband Stopband Transition MATLAB Design Ripple Ripple Band Command Butterworth No No Loose butter Chebyshev Yes No Tight cheby Chebyshev Type II (Inverse Chebyshev) No Yes Tight cheby Eliptic Yes Yes Tightest ellip ELEC 34: Systems April