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## Performance Analysis of Digital communication Systems

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**Performance Analysis of Digital communication Systems**DIGITAL COMMUNICATION**Block Diagram**Receiver Digital message Source Sink Transmitter Channel Functional model of digital communication system • : digital message. • :Transmitted signal. • : The channel noise. • : The received signal. • : An estimate of the transmitted digital message.**Performance Measure (Figure of Merit):**Waterfall shape • : i.e. The probability of error, it could be the Bit error Rate (BER) or Symbol Error Rate (SER). • : Energy per transmitted bit (Joule). • : Single sided PSD of White noise (Watt/Hz). • is dimensionless and equivalent to the**P(0/0)**“0” “0” Probability of Error: P(1/0) P(0/1) • : Probability of receiving “0” given that “0” was transmitted which is probability of correct. • : Probability of receiving “1” given that “1” was transmitted which is probability of correct. • : Probability of receiving “1” given that “0” was transmitted which is probability of error. • : Probability of receiving “0” given that “1” was transmitted which is probability of error. Rx Tx “1” “1” P(1/1) Binary Symmetric Channel**Probability of Error (Cont.):**• Probability of error ≡ Probability of Rx “0” given that “1” is Tx or vice versa • Pe = P(0) P(1/0) + P(1) P(0/1) • P(0) probability of sending “0”. • P(1) probability of sending “1”. • If P(0)=P(1) equally likely (equally probable). • Since the channel is symmetric**Channel Noise:**• is a sample function of which is assume to be Wide-sense Stationary process (WSS) with zero mean and Power spectral density (PSD): . Also its distribution is assumed to be Gaussian. • is called Additive White Gaussian Noise (AWGN). • The cannel is called also AWGN channel. Receiver Digital message Source Sink Transmitter Channel**Gaussian Distribution:**• (PDF) • (CDF)**G.D (Cont.)**area • Where is the CDF for a zero mean, unity variance G.R.V. • Let Called the Q-function.**Q-tables**• e.g.: Q(2.2)=?**G.D. (Cont.)**Area 1 0.5**G.D. (Cont.)**• The complementary error function is defined as: • erfc-function and the Q-function are related by:**Random Processes and LTI Systems:**LTI system Output Input WSS**R.P. & LTI Systems (Cont.)**• In our assumption the noise is Gaussian with , and , so the output from an LTI system is Gaussian with . • Since which is the average power of .**Thermal Noise in Communication Systems:**• A natural noise source is thermal noise, who amplitude statistics are well modeled to be Gaussian with zero mean. • The autocorrelation and PSD are well modeled as: • Where k = 1.38 × 10−23 (joule/Kelvin) is Boltzmann’s constant, G is conductance of the resistor (mhos); T is temperature in degrees Kelvin; and is the statistical average of time intervals between collisions of free electrons in the resistor (on the order of sec).**White Noise:**• The noise PSD is approximately flat over the frequency range of 0 to 10 GHz ⇒ let the spectrum be flat from 0 to ∞: • Where is a constant. • Noise that has a uniform spectrum over the entire frequency range is referred to as white noise. • The autocorrelation of white noise is: • Since for , any two different samples of white noise, no matter how close in time they are taken, are uncorrelated. • Since the noise samples of white noise are uncorrelated, if the noise is both white and Gaussian (for example, thermal noise) then the noise samples are also independent.**Performance Analysis of Baseband System with Polar line**Coding: p(t) Ap -p(t) “1” “0” Tp t t Tp To To - Ap “1” “1” “0” “1” s(t) Ap r(t) Tp To 2To 3To Decision Making Instants Ap+n<0 -Ap+n<0 Correct Decision Detection error - Ap 11011001**Rx**Tx Channel Threshold Detector Decision Making Device Sampler Optimum Threshold Let +A N: Noise amplitude at the sampling instant R.V. -Ap N: Gaussian R.V. with zero mean and variance (n) # - Ap Ap /**Cont.** P(error/1) = P(n < -Ap) = P(n > +Ap) = Q(Ap/) ) ) + ) Let Unipolar ?Bipolar ?**Pe can be minimized by maximizing (Ap/), this will be done**by passing the digital signal through an optimum filter (called match filter). Filter h(t) Threshold Detector The filter will enhance the pulse amplitude & at the same time reduce the noise power () t t To tm=To 2To**Cont.**After adding the filter we maximize**Cont.** Using Schwarz inequality: The equality holds if Let Equality if: ; If p(t) is real =**Cont.**(Unitless) Where Epis the energy of p(t) Match filter**p(t)**tm < To Non causal t t tm To To tm > To tm = To • Min. delay for • decision making t t tm To=tm To** tm = To**The optimum value The width of is 2To & symmetrical about To because: Multiplies both the signal and the noise • Does not affect = 1 Which is the energy**Alternative method to realize a match filter:**Filter • But hopt(t) Threshold Detector**Cont. (Correlator)**Integrate and dump Which is the time cross-correlation function. Threshold Detector Correlation Receiver :Correlator**Optimum linear receiver for a general binary system:**Filter • Let us transmit “1” as and “0” as . h(t) Threshold Detector Decision Making device is the Optimum Threshold**Cont.**• Let Which is a sampled version of is a G.R.V with zero mean & variance • Let since it is time independent. • is a G.R.V. with mean or depending on whether or “0” with variance .**Cont.**• The Conditional PDFs of are:**Cont.**Conditional PDFs y Equally likely**Cont.**= =**Cont.**• : Which is the arithmetic mean.**Cont.**• To minimize we have to maximize .**Cont.**• Using Schwarz inequality: where is an arbitrary constant.**Cont.**• For AWGN • from Parseval’s theorem • , since**Cont.**• Where: • . • .**Cont.**• Hints:**Receiver design**Threshold Detector Threshold Detector + + - - 0 0**Polar Signaling:**• “1” • “0” • & • Since noise & signal are multiplied by the same ratio**Polar(Cont.)**• is the average energy per bit. • ,**Polar(Cont.)**• Average Transmitted power. 1 10 15 (dB)**Unipolar (On-Off signaling)**• 1” • “0” • ,**Orthogonal Signals**• Orthogonality: • , , • For baseband Binary: orthogonal signals are not better than Polar.**Signal Space Analysis:**Receiver Digital message Source Sink Transmitter Channel • Message source emits one symbol every • Symbol belongs to an alphabet of M symbols • With a priori probabilities**Cont.**• @ • @Rx: