Signal Transmission Through LTI Systems EE 442 Spring 2017 Lecture 3. Signal Transmission

Transcription

1 Signal Transmission Through LTI Systems EE 442 Spring 207 Lecture 3 Signal Transmission

2 Steay-State Response in Linear Time Invariant Network By steay-state we mean an sinusoial excitation. x(t) LTI Network H(f) y(t) A sinusoial signal of frequency f at the input x(t) prouces a sinusoial signal of frequency f at the output y(t). The output y(t) Is given by y( t) H( f) x( t) y(t) will moify input x(t) by a change in magnitue an in phase. However, the frequency f will be unchange an the output will be causal. Signal Transmission 2

3 Pulse Response in Linear Time Invariant Network We are intereste in the pulse response in a given LTI system with a boune input boune output (BIBO). Lathi & Ding pp xt () X( f) LTI Network h(t) & H(f) y( t) x( t) h( t) Y( f ) X( f ) H( f ) where x(t) is the input, h(t) is impulse response of the network an y(t) is the output (Note: the symbol * enotes convolution). x( t) X( f ), h( t) H( f ) an y( t) Y( f ) where H( f) is the transfer function of the network. We can write H( f ) H( f ) e j ( f) j ( f) j( ( f ) j ( f )) an Y( f ) X( f ) H( f ) e y x h h e by convolution theorem Signal Transmission 3

4 Example: Pulse Response in a LTI Network xt () ht () This is special case of the transient response of a LTI network. Signal Transmission 4

5 Signal Distortion During Signal Transmission In amplifiers an transmission over a channel we want the output waveform to be a replica of the input waveform. This means we want istortionless transmission. Another way to say this: If x(t) is the input signal, then the output signal y(t) is require to be y(t) = K x(t-t ) (K is a constant) This means y(t) has it amplitue moifie by factor K an it is time shifte by time t. In the frequency omain we have: The Fourier transform is Y(f) = K X(f)e -j2ft by application of the convolution theorem. But we have not shown this yet! Signal Transmission 5

6 Signal Distortion During Signal Transmission For istortionless transmission the transfer function H(f) we can write, From H( f ) A e j2 ft H( f ) A an ( f ) 2 ft h H(f) is transfer function Conclusion: Distortionless transmission requires a constant amplitue H(f) over frequency an a linear phase response h (f) passing through the origin at f = 0. A H(f) f h (f) Signal Transmission 6

7 All-Pass System versus Distortionless Systems All-Pass System: Has a constant amplitue response, but oesn t have a linear phase response. A istortionless system is always an all-pass system, but the converse is not true in general. Transmission phase characteristic (if it oesn t have a constant slope) causes istortion. In practice, many LTI systems only approximate a linear phase response that passes through f = 0. Hence, t h( f) ( f) nees to have a constant slope. 2 f otherwise, the time elay t varies with frequency. For ES 442: phase istortion is important in igital communication systems because a nonlinear phase characteristic in a channel causes pulse ispersion (spreaing out) an causes pulses to interfere with ajacent pulses (calle interference). Signal Transmission 7

8 Signal Transmission 8

9 Time Delay of Ieal Transmission Line What is the time elay of a coax transmission line of length L? The elay varies linearly with the length of the transmission line. Z L = Z S = 50 L t (raians) (raians/sec) irection of travel t an L velocity 2 cycles = 4 raians elay shown x Signal Transmission 9

10 Phase Delay in Ieal Transmission Line Z L = Z S = 50 For an air-fille coaxial transmission line the time elay is roughly nanosecon (0-9 secon) per foot of physical length L. For a transmission line with a polyethylene ielectric the time elay is of the orer of.5 nanosecon per foot of length. L Linear Phase Response t Length = L 2 f Signal Transmission 0

11 Example (Lathi & Ding pp ) g(t) input R + + C output y(t) We have an low-pass filter, fin the transfer function H(f), an sketch H(f), the phase n (f) an elay t (f). The transfer function for y(f)/g(f) is ( ) a H f general form: where a j(2 f) a j(2 f ) Signal Transmission

12 Example (Lathi & Ding pp ) ( ) a H f general form: where a j(2 f) a j(2 f ) Therefore, the magnitue H( f ) an phase ( f ) are given by H( f), an 2 2 f 2 b 2 f h( f ) tan tan where b 2 f a 2 h( f ) f for 2 f (low frequency) a h Signal Transmission 2

13 Example (Lathi & Ding pp ) The time elay t is erivative of the phase with respect to frequency, hence, t h( f ) h( f ) h( f ) ( f), By efinition (2 f ) 2 f which is the slope of the phase versus frequency plot. For our low-pass filter example the time elay is t h ( f ) 2 f ( f) tan (2 f ) (2 f ) tan a From a table of erivatives: (tan x) x x 2 Signal Transmission 3

14 t Example (Lathi & Ding pp ) h( f ) 2 f ( f) tan (2 f ) (2 f ) 2 f t( f) f t( f) (2 f ) (2 f) Of course, for very low frequencies: 2 (2 f) 2 (2 f ) t( f) for 2 f a a Signal Transmission 4

15 Low-Pass Filter Example (Lathi & Ding) Plot from page 29 of Lathi & Ding: Amplitue response within 2% of peak value a Slope is - t (constant phase shift asymptote) Signal Transmission 5

16 Example (Lathi & Ding page 29) t ( ) f a a 0 a 2 f Signal Transmission 6

17 Ieal Filter versus Practical Filter The signal g(t) is transmitte without istortion, but has a time elay of t. Brick filter B f H( f ) an h( f ) 2 ft ; 2B f j2 ft H( f) e an 2B f j2 ft h t F e B B t t 2B ( ) 2 sinc 2 ( ) H( f ) ht () ( f ) 2 ft h Violates causality B f t Lathi & Ding; Page 30 2B t Signal Transmission 7

18 Ieal Filter versus Practical Filter II Impulse response h(t) is response to impulse (t) applie at time t = 0. The ieal filter on the previous slie is noncausal unrealizable. Practical approach to filter esign is to cutoff h(t) for t < 0. h ˆ( t) h( t) u( t) A goo approximation if t is large (t approaches infinity for ieal filter). Many ifferent practical non-ieal filters exist: Butterworth Chebyshev I Chebyshev II Elliptic Signal Transmission 8

19 Butterworth Filter (Maximally Flat Magnitue) Nt h or (n th orer filter) e r Signal Transmission 9

20 Butterworth Filter (Maximally Flat Magnitue) Im(s) Cauer topology Unit circle Re(s) H( j) 2 H 2 (0) C 2n Sallen Key topology Signal Transmission 20

21 Butterworth Filter Impulse Response Fourth-orer filter ht () H( f) h ( f ) Lathi & Ding; Page 33 Signal Transmission 2

22 Comparing Butterworth, Chebyshev & Bessel Filters H(f) H(f) h(t) Unit step response Unit impulse response Signal Transmission 22

23 Phase Delay versus Group (Envelope) Delay Phase response of H( f ) is ( f ), therefore h( f0 ) Phase elay ( at one frequency f ) is t, 0 2 f Group elay ( over a frequency ban) h t grp 0 an h( f) (2 f) Envelope Input: x( t) A( t) cos(2 ft ) Output: y( t) H( f ) A( t t ) cos(2 f ( t t ) ) grp Signal Transmission 23

24 Group (Envelope) Delay Group elay is an important way to escribe a filter's pass ban characteristics. Group elay is: () A measure of a network s phase istortion. (2) The transit time of a signal through a evice versus frequency. (3) The erivative of the evice's phase characteristic with respect to frequency (mathematical statement). Consier a simple example of a square wave, which as you know, is compose of a large group of frequency components. A square wave is square only because its frequency components are in proper phase alignment with one another. If we pass a square wave through a network an expect it to remain square, then we nee to ensure that the evice oesn't misalign these frequency components. Signal Transmission 24

25 Channel Impairments (Overview) Linear istortion cause by impulse response. y( t) h( t) g( t) Y( f ) H( f ) G( f ) H(f) attenuates an phase shifts the signal. Nonlinear istortion (e.g., such as from clipping) yt () x p xt () x p x() t x p x x() t x p x() t x p p Ranom Noise (inepenent or signal epenent) Interference from other transmissions or sources Self interference & ISI (from reflections or multipath) ISI is intersymbol interference Signal Transmission 25

26 Pulse Distortion in a Sine-Square Pulse Input pulse Output pulse This is typically what pulse istortion looks like in channels. (a) Amplitue-frequency istortion an phase-frequency istortion Input pulse Input pulse Output pulse Output pulse (b) Amplitue-frequency istortion (c) Phase-frequency istortion Signal Transmission 26