Frequency Response Analysis

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Marybeth Jennings
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1 Frequency Response Analysis

4 Lowpass Filter H( s) = ω c s + ω c H( jω ) = ω c jω + ω c Frequency Response Highpass Filter H( s) = H( jω ) = jω jω + ω c s s + ω c Bandpass Filter H( s) = H( jω ) = (Cascade Connection of Lowpass and Highpass) ω cb s s + ( ω ca + ω cb )s + ω ca ω cb jωω cb ( jω ) 2 + jω ω ca + ω cb ( ) + ω ca ω cb * M. J. Roberts - All Rights Reserved 4

6 Frequency Response Bandstop Filter H( s) = H( jω ) = s 2 + 2ω cb s + ω ca ω cb s 2 + ( ω ca + ω cb )s + ω ca ω cb ( jω ) 2 + j2ωω cb + ω ca ω cb jω ( ) 2 + jω ( ω ca + ω cb ) + ω ca ω cb * M. J. Roberts - All Rights Reserved 6

7 Frequency Response A biquadratic filter can be realized as a second-order system. Adjusting the parameter β changes the nature of the frequency response. It can emphasize or de-emphasize frequencies near its center frequency. * M. J. Roberts - All Rights Reserved 7

9 Ideal Filters Filters separate what is desired from what is not desired In the signals and systems context a filter separates signals in one frequency range from signals in another frequency range An ideal filter passes all signal power in its passband without distortion and completely blocks signal power outside its passband 12/29/10 M. J. Roberts - All Rights Reserved 9

10 Distortion Distortion is construed in signal analysis to mean changing the shape of a signal Multiplication of a signal by a constant (even a negative one) or shifting it in time do not change its shape No Distortion Distortion 12/29/10 M. J. Roberts - All Rights Reserved 10

11 Distortion Since a system can multiply by a constant or shift in time without distortion, a distortionless system would have an impulse response of the form ( ) h( t) = Aδ t t 0 The corresponding frequency response is H( f ) = Ae j 2π ft 0 12/29/10 M. J. Roberts - All Rights Reserved 11

14 Bandwidth Bandwidth generally means a range of frequencies This range could be the range of frequencies a filter passes or the range of frequencies present in a signal Bandwidth is traditionally construed to be range of frequencies in positive frequency space 12/29/10 M. J. Roberts - All Rights Reserved 14

17 Impulse Response and Causality All the impulse responses of ideal filters contain sinc functions, alone or in combinations, which are infinite in extent Therefore all ideal-filter impulse responses begin before time t = 0 This makes ideal filters non-causal Ideal filters cannot be physically realized, but they can be closely approximated 12/29/10 M. J. Roberts - All Rights Reserved 17

23 Noise Removal A very common use of filters is to remove noise from a signal. If the noise bandwidth is much greater than the signal bandwidth a large improvement in signal fidelity is possible. 12/29/10 M. J. Roberts - All Rights Reserved 23

24 The Decibel The bel ( B) ( named in honor of Alexander Graham Bell)is defined as the common logarithm (base 10) of a power ratio. So if the excitation of a system is X and the response is Y, the power gain of the system is P Y / P X. Expressed in bels that would be ( P Y / P X ) B = log 10 P Y / P X ( ) = log 10 Y 2 / X 2 ( ) ( ) = 2log 10 Y / X Since the prefix deci means one-tenth, that same power ratio expressed in decibels ( db)would be ( P Y / P X ) db = 10log 10 ( P Y / P X ) = 20log 10 ( Y / X) 12/29/10 M. J. Roberts - All Rights Reserved 24

25 The Decibel If a frequency response magnitude is the magnitude of the ratio of a system response to a system excitation ( ) = Y( jω ) X( jω ) H jω then that magnitude ratio, expressed in decibels, is Y( jω ) H( jω ) db = 20log 10 H( jω ) = 20log 10 X( jω ) = Y( jω ) X( jω ) db db 12/29/10 M. J. Roberts - All Rights Reserved 25

26 Log-Magnitude Frequency- Response Plots Consider the two (different) transfer functions, H 1 ( jω ) = 1 jω +1 and H 2 ( jω ) = ω 2 + j31ω When plotted on this scale, these magnitude frequency response plots are indistinguishable. 12/29/10 M. J. Roberts - All Rights Reserved 26

28 Bode Diagrams A magnitude-frequency-response Bode diagram is a graph of the frequency response magnitude in db against a logarithmic frequency scale. H 1 H 2 ( jω ) = ( jω ) = 1 jω ω 2 + j31ω 12/29/10 M. J. Roberts - All Rights Reserved 28

29 Bode Diagrams Continuous-time LTI systems are described by equations of the general form, N d k a k dt y( t) M d k = b k k k=0 k=0 The corresonding transfer function is ( ) dt k x t H s ( ) = b M s M + b M 1 s M b 1 s + b 0 a N s N + a N 1 s N a 1 s + b 0 12/29/10 M. J. Roberts - All Rights Reserved 29

30 Bode Diagrams The transfer function can be written in the form ( H( s) = A 1 s / z 1) ( 1 s / z 2 ) ( 1 s / z M ) ( 1 s / p 1 )( 1 s / p 2 ) ( 1 s / p N ) where the z s are the values of s at which the frequency response goes to zero and the p s are the values of s at which the frequency response goes to infinity. These z s and p s are commonly referred to as the zeros and poles of the system. The frequency response is ( H( jω ) = A 1 jω / z 1) ( 1 jω / z 2 ) ( 1 jω / z M ) ( 1 jω / p 1 )( 1 jω / p 2 ) ( 1 jω / p N ) 12/29/10 M. J. Roberts - All Rights Reserved 30

31 Bode Diagrams From the factored form of the frequency response a system can be conceived as the cascade of simple systems, each of which has only one numerator factor or one denominator factor. Since the Bode diagram is logarithmic, multiplied frequency responses add when expressed in db. 12/29/10 M. J. Roberts - All Rights Reserved 31

32 Bode Diagrams System Bode diagrams are formed by adding the Bode diagrams of the simple systems which are in cascade. Each simple-system diagram is called a component diagram. One Real Pole H( jω ) = 1 1 jω / p k 12/29/10 M. J. Roberts - All Rights Reserved 32

33 Bode Diagrams Let the frequency response of a lowpass filter be H( jω ) = 1 j ω +1 This can be written as H( jω ) = 1 1 jω 20, 000 ( ) Its Bode diagram has one corner frequency at ω = 20, /29/10 M. J. Roberts - All Rights Reserved 33

38 Complex Pole Pair H( jω ) = Bode Diagrams 1 jω p jω p 2 = 1 ( ) ( )2 1 jω 2 Re p jω p 1 p 1 2 The natural radian frequency ω n is defined by ω n 2 = p 1 p 2 The damping ratio ζ is defined by ζ = p 1 + p 2 2 p 1 p 2 12/29/10 M. J. Roberts - All Rights Reserved 38

42 Practical Active Filters Operational Amplifiers The ideal operational amplifier has infinite input impedance, zero output impedance, infinite gain and infinite bandwidth. ( ) = V ( s) o ( s) = Z f ( s) ( s) H s V i Z i ( ) = Z f ( s) + Z i ( s) ( s) H s Z i 12/29/10 M. J. Roberts - All Rights Reserved 42

47 Distortion Distortion means the same thing for discrete-time signals as it does for continuous-time signals, changing the shape of a signal No Distortion Distortion 12/29/10 M. J. Roberts - All Rights Reserved 47

48 Distortion A distortionless system would have an impulse response of the form, [ ] = Aδ n n 0 h n [ ] The corresponding transfer function is H( e jω ) = Ae jωn 0 12/29/10 M. J. Roberts - All Rights Reserved 48

Two-Dimensional Filtering of Images Causal

Causal Lowpass Filtering of Columns in an

Two-Dimensional Filtering of Images Non-Causal Lowpass Filtering of Rows in an Image

59 Two-Dimensional Filtering of Images Causal Lowpass Filtering of Rows and Columns in an Image Non-Causal Lowpass Filtering of Rows and Columns in an Image 12/29/10 M. J. Roberts - All Rights Reserved 59

65 Discrete-Time Filters Moving-Average Filter H e jω ( )Ω/2 ( ) = e j N 1 N ( ) sin( Ω / 2) sin NΩ / 2 ( ) = e j ( N 1)Ω/2 drcl Ω / 2π, N ( ) / N h n = u n u n N Always Stable 12/29/10 M. J. Roberts - All Rights Reserved 65

66 Discrete-Time Filters Ideal Lowpass Filter Impulse Response Almost-Ideal Lowpass Filter Impulse Response Almost-Ideal Lowpass Filter Magnitude Frequency Response 12/29/10 M. J. Roberts - All Rights Reserved 66

68 Advantages of Discrete-Time Filters They are almost insensitive to environmental effects Continuous-time filters at low frequencies may require very large components, discrete-time filters do not Discrete-time filters are often programmable making them easy to modify Discrete-time signals can be stored indefinitely on magnetic media, stored continuous-time signals degrade over time Discrete-time filters can handle multiple signals by multiplexing them 12/29/10 M. J. Roberts - All Rights Reserved 68

69 Due date: Oct. 28, 2016

Fourier Transform Analysis of Signals and Systems

Fourier Transform Analysis of Signals and Systems Ideal Filters Filters separate what is desired from what is not desired In the signals and systems context a filter separates signals in one frequency