1. ECE 4710: Lecture #5 1 Linear Systems Linear System Input Signal x(t) Output Signal y(t) h(t)  H( f ) Voltage Spectrum (via FT) AutoCorrelation Function Power Spectral Density Useful Signal Characterizations

2. ECE 4710: Lecture #5 2 Linear Systems  Linear Time Invariant (LTI) Conditions:  Linear  Superposition holds  Time Invariant  Shape of system response, H( f ), is same no matter when input is applied to system »Does not apply for most mobile (wireless) communication channels  Impulse Response = h(t)  h(t) = 0 for t < 0  causal  y(t) = h(t) when x(t) =  (t)  delta impulse function at input can measure system response

3. ECE 4710: Lecture #5 3 System Output  Output is result of convolution integral between input and impulse response function  Convolution integral is difficult to evaluate  Preferred approach is to find system transfer function H( f )  Convolution in time is multiplication in frequency

4. ECE 4710: Lecture #5 4 System Transfer Function  H( f ) is the FT of h(t) and is called the frequency response or the transfer function of the system  In general H( f ) is a complex function with magnitude and phase response:  Magnitude response is even function in frequency  Positive and negative frequencies have same amplitude  Phase response is an odd function in frequency

5. ECE 4710: Lecture #5 5 System Transfer Function  H( f ) can be measured by using sinusoidal test input signal and sweeping the frequency over the desired range  Spectrum analyzer uses this approach  How is power content of input signal affected by the system?

6. ECE 4710: Lecture #5 6 Power Transfer  Power Transfer Function  Example: RC Low Pass Filter (LPF)  Find G h ( f ) + x(t)  + y(t)  R C i(t)

7. ECE 4710: Lecture #5 7 RC LPF  KVL around loop:  Capacitor current related to voltage drop:  Table 2-1, pg. 52 :  Take FT of both sides:  Solving for transfer function:

8. ECE 4710: Lecture #5 8 RC LPF  Table 2-2, pg. 64:  So where  = RC is the time constant of the LPF 1 RC t e -1 RC  = RC h(t) = Impulse Response Function of ILPF

9. ECE 4710: Lecture #5 9 RC LPF  Power Transfer Function:  Define f o = cutoff frequency = 1 / 2  RC so  At f = f o  G h ( f o ) = 0.5  Power @ f o attenuated by half  half power or 3 dB BW

10. ECE 4710: Lecture #5 10 Distortionless Transmission  Distortionless channel is very desirable in a communication system  Output is simply delayed replica of input : y(t) = A x (t - T d ) where A : channel loss ( A < 1) T d : time delay in channel  In frequency domain a distortion free response is  Thus, a distortion free channel has Note that there is no frequency dependence for amplitude but there is for phase

11. ECE 4710: Lecture #5 11 Distortionless Transmission  LTI system will have no distortion if 1) Amplitude response is flat  2) Phase response is linear function of frequency   Distortion classified as either 1) amplitude or 2) phase distortion  Looking at phase distortion from time delay standpoint:  Time delay of channel/system must be independent of frequency, otherwise phase distortion will occur

12. ECE 4710: Lecture #5 12 RC Filter Distortion  Filter Transfer Function:  Amplitude response:  Phase Response:  Time Delay:  Not a constant value

13. ECE 4710: Lecture #5 13 For f < 0.5 f o the amplitude distortion is < 0.5 dB (~12%) RC Amplitude Distortion

14. ECE 4710: Lecture #5 14 For f < 0.5 f o the phase distortion is < 2.1  (~8%) RC Phase Distortion

15. ECE 4710: Lecture #5 15 High frequency signal components have less delay For f o = 1 kHz  delay is ~0.2 msec RC Time Delay

16. ECE 4710: Lecture #5 16 Distortion  Most communication systems have both amplitude and phase distortion of H( f )  Distortion can be acceptable depending on  Distortion type  amplitude, phase, or both  Distortion magnitude  e.g. how bad?  Type of information signal  audio, video, or data

17. ECE 4710: Lecture #5 17 Audio Distortion  Human ear is sensitive to amplitude distortion in f  Audio spectrum is 300 Hz to 15 kHz  15  phase distortion causes time delay of ~ 3  sec  Duration of spoken syllable is 10-100 msec  3  sec delay is imperceptible (< 0.05 % of 10 msec syllable)  3 dB amplitude error is very noticeable to ear  High-fidelity audio amplifiers  Focus on maintaining flat spectral response  Phase distortion is not concern

18. ECE 4710: Lecture #5 18 Video Distortion  For analog video transmission the phase distortion is the primary concern  Amplitude variations will cause variations in image intensity  Phase variations will cause time delays which will cause objects in the image to blur at edges  Human eye is more sensitive to phase variations  Analog video filters require excellent phase linearity

19. ECE 4710: Lecture #5 19 Data Distortion  For digital data both amplitude and phase distortion can have serious affects  Rectangular data pulse train:  Pulse smearing into adjacent symbol time slots  Inter-Symbol Interference (ISI)  Increase probability that bit errors will occur (BER  )  Special filters designed for digital data to minimize impact of ISI 0 1 0 1 0 TsTs Bandlimited Communication System or Channel 0 1 0 1 0