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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 )
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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 ?
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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
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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 !!
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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
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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.)
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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
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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
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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
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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”
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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”
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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
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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
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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
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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!!
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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 :
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