1 Dr. Uri Mahlab

INTRODUCTION In order to transmit digital information over * bandpass channels, we have to transfer the information to a carrier wave of.appropriate frequency We will study some of the most commonly * used digital modulation techniques wherein the digital information modifies the amplitude the phase, or the frequency of the carrier in.discrete steps 2 Dr. Uri Mahlab

The modulation waveforms for transmitting :binary information over bandpass channels ASK FSK PSK DSB 3 Dr. Uri Mahlab

OPTIMUM RECEIVER FOR BINARY :DIGITAL MODULATION SCHEMS The function of a receiver in a binary communication * system is to distinguish between two transmitted signals.S 1 (t) and S 2 (t) in the presence of noise The performance of the receiver is usually measured * in terms of the probability of error and the receiver is said to be optimum if it yields the minimum.probability of error In this section, we will derive the structure of an optimum * receiver that can be used for demodulating binary.ASK,PSK,and FSK signals 4 Dr. Uri Mahlab

Description of binary ASK,PSK, and : FSK schemes -Bandpass binary data transmission system Modulator Channel (H c (f Demodulator (receiver) {b k } Binary data Input {bk}{bk} Transmit carrier Clock pulses Noise (n(t Clock pulses Local carrier Binary data output (Z(t + + (V(t ּ+ּ+ 5 Dr. Uri Mahlab

:Explanation * The input of the system is a binary bit sequence {b k } with a *.bit rate r b and bit duration T b The output of the modulator during the Kth bit interval *.depends on the Kth input bit b k The modulator output Z(t) during the Kth bit interval is * a shifted version of one of two basic waveforms S 1 (t) or S 2 (t) and :Z(t) is a random process defined by.1 6 Dr. Uri Mahlab

The waveforms S 1 (t) and S 2 (t) have a duration * of T b and have finite energy,that is,S 1 (t) and S 2 (t) =0 if and Energy :Term 7 Dr. Uri Mahlab

:The received signal + noise 8 Dr. Uri Mahlab

Choice of signaling waveforms for various types of digital* modulation schemes S 1 (t),S 2 (t)=0 for.The frequency of the carrier f c is assumed to be a multiple of r b Type of modulation ASK PSK FSK 0 9 Dr. Uri Mahlab

:Receiver structure Threshold device or A/D converter (V 0 (t Filter (H(f output Sample every T b seconds 10 Dr. Uri Mahlab

:{Probability of Error-{P e* The measure of performance used for comparing * !!!digital modulation schemes is the probability of error The receiver makes errors in the decoding process * !!! due to the noise present at its input The receiver parameters as H(f) and threshold setting are * !!!chosen to minimize the probability of error 11 Dr. Uri Mahlab

:The output of the filter at t=kT b can be written as * 12 Dr. Uri Mahlab

:The signal component in the output at t=kT b h( ) is the impulse response of the receiver filter* ISI=0* 13 Dr. Uri Mahlab

Substituting Z(t) from equation 1 and making* change of the variable, the signal component :will look like that 14 Dr. Uri Mahlab

:The noise component n 0 (kT b ) is given by *.The output noise n 0 (t) is a stationary zero mean Gaussian random process :The variance of n 0 (t) is* :The probability density function of n 0 (t) is* 15

The probability that the kth bit is incorrectly decoded* :is given by.2 16 Dr. Uri Mahlab

:The conditional pdf of V 0 given b k = 0 is given by* :It is similarly when b k is 1*.3 17 Dr. Uri Mahlab

Combining equation 2 and 3, we obtain an* :expression for the probability of error- P e as.4 18 Dr. Uri Mahlab

:Conditional pdf of V 0 given b k :The optimum value of the threshold T 0 * is* 19 Dr. Uri Mahlab

Substituting the value of T* 0 for T 0 in equation 4* we can rewrite the expression for the probability :of error as 20 Dr. Uri Mahlab

The optimum filter is the filter that maximizes* the ratio or the square of the ratio (maximizing eliminates the requirement S 01 <S 02 ) 21 Dr. Uri Mahlab

:Transfer Function of the Optimum Filter* The probability of error is minimized by an * appropriate choice of h(t) which maximizes Where And 22 Dr. Uri Mahlab

If we let P(t) =S 2 (t)-S 1 (t), then the numerator of the* :quantity to be maximized is Since P(t)=0 for t<0 and h( )=0 for <0* :the Fourier transform of P 0 is 23 Dr. Uri Mahlab

:Hence can be written as* (*) We can maximize by applying Schwarz’s* :inequality which has the form (**) 24 Dr. Uri Mahlab

Applying Schwarz’s inequality to Equation(**) with- and We see that H(f), which maximizes,is given by- !!! Where K is an arbitrary constant (***) 25 Dr. Uri Mahlab

Substituting equation (***) in(*), we obtain- :the maximum value of as :And the minimum probability of error is given by- 26 Dr. Uri Mahlab

:Matched Filter Receiver* If the channel noise is white, that is, G n (f)= /2,then the transfer - :function of the optimum receiver is given by From Equation (***) with the arbitrary constant K set equal to /2- :The impulse response of the optimum filter is 27 Dr. Uri Mahlab

Recognizing the fact that the inverse Fourier * of P*(f) is P(-t) and that exp(-2 jfT b ) represent :a delay of T b we obtain h(t) as :Since p(t)=S 1 (t)-S 2 (t), we have* The impulse response h(t) is matched to the signal * :S 1 (t) and S 2 (t) and for this reason the filter is called MATCHED FILTER 28 Dr. Uri Mahlab

:Impulse response of the Matched Filter * (S 2 (t (S 1 (t 2 \T b TbTb t t t t t (a) (b) (c) 2 \T b (P(t)=S 2 (t)-S 1 (t (P(-t T b- 0 2 (d) 2 \T b 0 TbTb (h(T b -t)=p(t 2 (e) (h(t)=p(T b -t 29 Dr. Uri Mahlab

:Correlation Receiver* The output of the receiver at t=T b* Where V( ) is the noisy input to the receiver Substituting and noting * : that we can rewrite the preceding expression as (# #) 30 Dr. Uri Mahlab

Equation(# #) suggested that the optimum receiver can be implemented * as shown in Figure 1.This form of the receiver is called A Correlation Receiver Threshold device (A\D) integrator - + Sample every T b seconds Figure 1 31 Dr. Uri Mahlab

In actual practice, the receiver shown in Figure 1 is actually *.implemented as shown in Figure 2 In this implementation, the integrator has to be reset at the - (end of each signaling interval in order to ovoid (I.S.I !!! Inter symbol interference :Integrate and dump correlation receiver Filter to limit noise power Threshold device (A/D) R (Signal z(t + (n(t + White Gaussian noise High gain amplifier Closed every T b seconds c Figure 2 The bandwidth of the filter preceding the integrator is assumed * !!! to be wide enough to pass z(t) without distortion 32

Example: A band pass data transmission scheme uses a PSK signaling scheme with The carrier amplitude at the receiver input is 1 mvolt and the psd of the A.W.G.N at input is watt/Hz. Assume that an ideal correlation receiver is used. Calculate the.average bit error rate of the receiver 33 Dr. Uri Mahlab

:Solution 34 Dr. Uri Mahlab

=Probability of error = Pe * :Solution Continue 35 Dr. Uri Mahlab

* Binary ASK signaling schemes: The binary ASK waveform can be described as Where and We can represent :Z(t) as 36 Dr. Uri Mahlab

Where D(t) is a lowpass pulse waveform consisting of.rectangular pulses :The model for D(t) is 37 Dr. Uri Mahlab

:The power spectral density is given by The autocorrelation function and the power spectral density :is given by 38 Dr. Uri Mahlab

:The psd of Z(t) is given by 39 Dr. Uri Mahlab

If we use a pulse waveform D(t) in which the individual pulses g(t) have the shape 40 Dr. Uri Mahlab

Coherent ASK We start with The signal components of the receiver output at the :of a signaling interval are 41 Dr. Uri Mahlab

:The optimum threshold setting in the receiver is :The probability of error can be computed as 42 Dr. Uri Mahlab

:The average signal power at the receiver input is given by We can express the probability of error in terms of the :average signal power The probability of error is sometimes expressed in * : terms of the average signal energy per bit, as 43 Dr. Uri Mahlab

Noncoherent ASK :The input to the receiver is * 44 Dr. Uri Mahlab

Noncoharent ASK Receiver 45

:The pdf is 46 Dr. Uri Mahlab

pdf’s of the envelope of the noise and the signal * :pulse noise 47 Dr. Uri Mahlab

:The probability of error is given by 48 Dr. Uri Mahlab

49 Dr. Uri Mahlab

BINERY PSK SIGNALING SCHEMES :The waveforms are * :The binary PSK waveform Z(t) can be described by *.D(t) - random binary waveform * 50 Dr. Uri Mahlab

:The power spectral density of PSK signal is 51 Dr. Uri Mahlab

Coherent PSK :The signal components of the receiver output are 52 Dr. Uri Mahlab

:The probability of error is given by 53 Dr. Uri Mahlab

54 Dr. Uri Mahlab

DELAY LOGIC NETWORK LEVEL SHIFT BINERY SEQUENCE Z(t) DIFFERENTIALLY COHERENT * :PSK DPSK modulator 55 Dr. Uri Mahlab

DPSK demodulator Filter to limit noise power Delay Lowpass filter or integrator Threshold device (A/D) Z(t) 56 Dr. Uri Mahlab

Differential encoding & decoding 57 Dr. Uri Mahlab

* BINARY FSK SIGNALING SCHEMES : :The waveforms of FSK signaling :Mathematically it can be represented as 58 Dr. Uri Mahlab

Power spectral density of FSK signals Power spectral density of a binary FSK signal with 59 Dr. Uri Mahlab

Coherent FSK :The local carrier signal required is The input to the A/D converter at sampling time 60 Dr. Uri Mahlab

The probability of error for the correlation receiver is :given by 61 Dr. Uri Mahlab

.Which are usually encountered in practical system :We now have 62 :When Dr. Uri Mahlab

Noncoherent FSK 63 Dr. Uri Mahlab

Noncoharenr demodulator of binary FSK ENVELOPE DETECTOR ENVELOPE DETECTOR THRESHOLD DEVICE (A/D) + - Z(t)+n(t) 64 Dr. Uri Mahlab

Probability of error for binary digital modulation * :schemes 65 Dr. Uri Mahlab

M-ARY SIGNALING SCHEMES :M-ARY coherent PSK The M possible signals that would be transmitted :during each signaling interval of duration Ts are :The digital M-ary PSK waveform can be represented 66 Dr. Uri Mahlab

:In four-phase PSK (QPSK), the waveform are 67 Dr. Uri Mahlab

Phasor diagram for QPSK That are derived from a coherent local carrier reference 68

If we assume that S 1 was the transmitted signal :during the signaling interval (0,T s ),then we have 69 Dr. Uri Mahlab

Z(t) QPSK receiver scheme 70 Dr. Uri Mahlab

:The outputs of the correlators at time t=T S are 71 Dr. Uri Mahlab

Probability of error of QPSK: 72 Dr. Uri Mahlab

73 Dr. Uri Mahlab

Phasor diagram for M-ary PSK ; M=8 74 Dr. Uri Mahlab

The average power requirement of a binary PSK :scheme are given by 75 Dr. Uri Mahlab

* COMPARISION OF POWER-BANDWIDTH :FOR M-ARY PSK Value of M dB 3.91 dB 8.52 dB dB 76 Dr. Uri Mahlab

* M-ary for four-phase Differential PSK: RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK Integrate and dump filter Integrate and dump filter Z(t) 77 Dr. Uri Mahlab

:The probability of error in M-ary differential PSK :The differential PSK waveform is 78 Dr. Uri Mahlab

:Transmitter for differential PSK* Serial to parallel converter Diff phase mod. Envelope modulator BPF (Z(t Clock signal 2400 Hz 600 Hz 79 Dr. Uri Mahlab

* M-ary Wideband FSK Schemas: Let us consider an FSK scheme witch have the : following properties 80 Dr. Uri Mahlab

:Orthogonal Wideband FSK receiver MAXIMUM SELECTOR Z(t) 81 Dr. Uri Mahlab

:The filter outputs are 82 Dr. Uri Mahlab

:N 0 is given by :The probability of correct decoding as :In the preceding step we made use of the identity 83 Dr. Uri Mahlab

The joint pdf of Y2,Y3,…,YM * :is given by 84 Dr. Uri Mahlab

where 85 Dr. Uri Mahlab

Probability of error for M-ary orthogonal * : signaling scheme 86 Dr. Uri Mahlab

The probability that the receiver incorrectly * decoded the incoming signal S 1 (t) is P e1 = 1-P e1 The probability that the receiver makes * an error in decoding is P e = P e1 We assume that, and We can see that increasing values of M lead to smaller power requirements and also to more complex transmitting receiving equipment. 87 Dr. Uri Mahlab

In the limiting case as M the probability of error P e satisfies The maximum errorless r b at W data can be transmitted using an M- ary orthogonal FSK signaling scheme The bandwidth of the signal set as M 88 Dr. Uri Mahlab