1. ECE 4710: Lecture #31 1 System Performance  Chapter 7: Performance of Communication Systems Corrupted by Noise  Important Practical Considerations:  Complexity vs. Cost  Coherent vs. Non-Coherent Detection  Important Performance Measures:  Signal BW  Spectral Efficiency  Probability of Bit Error  P e or BER  Required S / N for given P e  digital systems only »Analog systems  output S / N only (no P e )

2. ECE 4710: Lecture #31 2 System Performance  Shannon’s Channel Capacity  Defines S / N & spectral efficiency for specific P e  Example: a digital modulation method with a S / N = 10 dB yields a 3 bps/Hz spectral efficiency @ P e = 10 -5  For a received signal corrupted by noise (channel + system) how do we determine the specific P e for a given S / N ?

3. ECE 4710: Lecture #31 3 System Performance  Numerous methods for signal demodulation and detection  Coherent vs. Non-Coherent  Optimum vs. Sub-optimum  Optimum  Maximize S / N and minimize P e »Usually coherent demodulation + specialized filtering/processing  Sub-optimum »Often done in order to lower cost  practical consideration  Non-coherent Rx has simpler circuitry »Sometimes performance is very close to optimum Rx for practical systems

4. ECE 4710: Lecture #31 4 Binary System Bandpass SuperH LNA, Mixer, IF, IF Filter + Amp, Detection, etc. Bit Synch Binary Decision / Detection Noise causes bit errors to occur !!

5. ECE 4710: Lecture #31 5 BER Evaluation  Develop general technique for determining Bit Error Rate (BER) for binary signaling  Transmitted bandpass (RF) signal over bit period T is  Baseband output signal (after RF/IF processing circuits) is  Baseband analog signal is corrupted by noise

6. ECE 4710: Lecture #31 6 BER Evaluation  Baseband analog waveform is sampled at some time t o during bit interval:  For matched filter processing circuits t o is usually t o = T »End of bit period  integration operation to average out signal fluctuations and reduce impact of noise  For simple processing t o is usually t o = T/2 »Middle of bit period  maximum eye opening of line code  is a random variable whose probability density function (PDF) is continuous b/c the signal is corrupted by noise (channel, system, etc.)

7. ECE 4710: Lecture #31 7 BER Evaluation  For simplified notation let so  is called the “test statistic” »Random variable with continuous PDF  Probability Density Function  PDF  Statistical characterization of random variation  For our purposes it is the random variation of received signal (which contains noise) at sampling point t 0

8. ECE 4710: Lecture #31 8 PDF  Received signal + noise over one bit period  PDF is ensemble average of r 0 (t 0 ) values Avg Signal Strength Noise Variation Avg Signal Strength Noise Variation Avg Signal Strength Noise Variation Avg Signal Strength Noise Variation

9. ECE 4710: Lecture #31 9 PDFs  Two PDFs  one for each possible state, r 01 or r 02, of received signal  Conditional PDFs  depend on transmitted state  Denote conditional PDFs as:  Functional shape of PDF depends largely on »Channel noise characteristics »Type of detector & filter circuits

10. ECE 4710: Lecture #31 10 Gaussian PDFs Must set threshold voltage V T to detect binary data r 0 > V T  “1” r 0 < V T  “0” Detection Decision : Binary “1” Binary “0”

11. ECE 4710: Lecture #31 11 Bit Errors  Signal + Noise at Rx  Errors occur in two ways for binary system:  If binary 1 is sent but  If binary 0 is sent but  Probability of error is integration of conditional PDF over “tail regions”  If binary 1 is sent   If binary 0 is sent  r 0 > V T  “1” r 0 < V T  “0”

12. ECE 4710: Lecture #31 12 Bit Error Rate  The rate of bit errors is the summation of the error type multiplied by the probability of the bit state  General expression for binary system  & are source statistics  Most applications assume all states are equally likely  For binary system then

13. ECE 4710: Lecture #31 13 Gaussian Noise  Shape of conditional PDFs depends on  Channel noise characteristics  Type of detector & filter circuits  In the absence of interference from other signals the channel noise typically has a Gaussian distribution  Channel noise is Additive White Gaussian Noise (AWGN) »Gaussian random noise process n(t) has flat PSD »“White light”  all colors of visible spectrum present »“White noise”  all frequencies (< B ) present in noise process

14. ECE 4710: Lecture #31 14 AWGN  Channel noise is typically (not always) AWGN for wireless communication systems when no interference is present  Not necessarily true for wired communication systems  Rx circuit acts upon input channel noise  Baseband noise will be AWGN if the Rx is linear (excluding threshold comparator) »SuperH with LNA, mixer, IF stage, & product detector can be linear »Not linear for Rx circuits with AGC, power limiters, non-linear detectors (envelope detector), etc

15. ECE 4710: Lecture #31 15 AWGN  For AWGN channel noise + linear Rx circuit the sampled baseband binary signal is where  s 01 & s 02  known constants for given Rx type and known input signal waveforms s 1 (t) and s 2 (t) Additive Noise!!

16. ECE 4710: Lecture #31 16 Sampled Output  Baseband noise is zero-mean Gaussian random variable  Sampled output r 0 is a Gaussian random variable with a mean value of either s 01 or s 02 depending on whether binary 1 or binary 0 is sent  Gaussian function :