DIGITAL CARRIER MODULATION SCHEMES Dr.Uri Mahlab 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 The modulation scheme using bandpass pulse ,shaping followed by analog modulation DSB or VSB- requires the minimum transmission bandwidth. However ,the equipment required to generate , transmit ,and demodulate the waveform is .quite complex 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 S1(t) and S2(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 Transmit carrier Local carrier Noise n(t) Clock pulses Clock pulses + Input Modulator Channel Hc(f) ּ+ Demodulator (receiver) Binary data + V(t) Z(t) {bk} Binary data output {bk} 5 Dr. Uri Mahlab

:Explanation The input of the system is a binary bit sequence {bk} with a .bit rate r b and bit duration Tb The output of the modulator during the Kth bit interval .depends on the Kth input bit bk The modulator output Z(t) during the Kth bit interval is a shifted version of one of two basic waveforms S1(t) or S2(t) and :Z(t) is a random process defined by 6 Dr. Uri Mahlab

The waveforms S1(t) and S2(t) have a duration * of Tb and have finite energy,that is, S1(t) and S2(t) =0 if and Energy :Term 7 Dr. Uri Mahlab

:The received signal + noise The shape of the waveforms depends on the type of the modulation used .The output of the modulator passes through a bandpass channel Hc(f) which for purposes of analysis is assumed to be an ideal channel with adequate bandwidth so the signal passes through without suffering any distortion other then propagation delay .the channel nosie n(t) is assumed to be a zero mean stationary, Gaussian random process with a known power spectral .(density Gn(f 8 Dr. Uri Mahlab

Type of modulation ASK PSK FSK 9 Choice of signaling waveforms for various types of digital* modulation schemes S1(t),S2(t)=0 for .The frequency of the carrier fc is assumed to be a multiple of rb Type of modulation ASK PSK FSK 9 Dr. Uri Mahlab

:Receiver structure V0(t) Threshold Filter device or A/D Hr(f) converter Filter Hr(f) output Sample every Tb seconds The receiver has to determine which of the two known waveforms s1(t) or .s2(t) was present at its input during each signaling interval The sampled value is compared against a predetermined threshold value T0 and the transmitted bit is decoded (with occasional errors) as 1 or 0 .depending on whether V0(kTb) is greater or less then threshold T0 10 Dr. Uri Mahlab

:{Probability of Error-Pe*} 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 We will assume that bk is an equiprobable,independent sequence * of bits.the accurrence of s1(t)or s2(t) during a bit interval dose not influence the occurrrnce of s1(t) or s2(t) during any other non-overlapping bit interval ;further , s1(t) and s2(t) are .equiprobable The channel noise will be assumed to be a zero mean Gaussian* .(random process with a power spectral density Gn(f .We will assume that the ISI generated by the filter is minimum* 11 Dr. Uri Mahlab

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

:The signal component in the output at t=kTb 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 n0(kTb) is given by .The output noise n0(t) is a stationary zero mean Gaussian random process The variance of n0(t) is The probability density function of n0(t) is 15 Dr. Uri Mahlab

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

The conditional pdf of V0 given bk = 0 is given by .3 :It is similarly when bk is 1 17 Dr. Uri Mahlab

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

Conditional pdf of V0 given bk The optimum value of the threshold T0* is 19 Dr. Uri Mahlab

Substituting the value of T*0 for T0 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 S01<S02) S01,S02 and depend on the choice of the filter impulse response or the transfer function 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 The receiver has to determine which of the two known waveforms* .s1(t) and s2(t) was present at its input during each signaling interval The optimum receiver distinguishes between s1(t) and s2(t) from the !noisy versions of s1(t) and s2(t) with minimum probability of error And 22 Dr. Uri Mahlab

If we let P(t) =S2(t)-S1(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 P0 is 23 Dr. Uri Mahlab

(*) (**) :Hence can be written as We can maximize by applying Schwarz’s* :inequality which has the form Where X1(f) and X2(f) are arbitrary complex functions of a common variable f. The equal sign in ** applies when X1(f)=KX*2(f) ,where K is an arbitrary constant and X*2(f) is .(the complex conjugate of X2(f (**) 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

Exrecises 27

Matched Filter Receiver Exrecise - 1 Matched Filter Receiver

Exrecise - 1 27 Matched Filter Receiver If the channel noise is white, that is, Gn(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 jfTb) represent a delay of Tb we obtain h(t) as Since p(t)=S2(t)-S1(t) , we have The impulse response h(t) is matched to the signal S1(t) and S2(t) and for this reason the filter is called MATCHED FILTER 28 Dr. Uri Mahlab

Impulse response of the Matched Filter S2(t) 1 t 2 \Tb (a) S1(t) 2 \Tb t 1- (b) 2 P(t)=S2(t)-S1(t) 2 \Tb t Tb 2 (c) (a).S1(t) (b).S2(t) (c).p(t)=S1(t)-S2(t) (d).p(-t) (e).h(t)=p(Tb-t) (P(-t t (d) Tb- 2 h(Tb-t)=p(t) h(t)=p(Tb-t) 2 \Tb t 29 (e) Tb Dr. Uri Mahlab

Exrecise - 2 Correlation Receiver

Exrecise - 2 Correlation Receiver The output of the receiver at t=Tb* Where V( ) is the noisy input to the receiver Substituting and noting that we can rewrite the preceding expression as (# #) 30 Dr. Uri Mahlab

A Correlation Receiver Equation(# #) suggested that the optimum receiver can be implemented as shown in Figure 1 .This form of the receiver is called A Correlation Receiver integrator Figure 1 Threshold device (A\D) - + It must be pointed out that Equation # # and the receiver* shown in Figure 1 require that the integration operation .be ideal with zero initial conditions Sample every Tb seconds integrator 31 Dr. Uri Mahlab

!!! Inter symbol interference 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 White Gaussian noise Closed every Tb seconds (n(t Filter to limit noise power Threshold device (A/D) + + c R (Signal z(t If RC >>Tb ,the circuit shown in Figure 2 very closely approximates an ideal integrator and operates with the same probability of error as the ideal receiver shown .in Figure 1 The sampling and discharging of the capacitor . dumping) must be carefully synchronized Furthermore ,the local reference signal S1(t)-S2(t) must be in “phase” with the signal component at the receiver input,that is, the !correlation receiver performs coherent demodulation High gain amplifier Figure 2 The bandwidth of the filter preceding the integrator is assumed to be wide enough to pass z(t) without distortion 32 Dr. Uri Mahlab

Exrecise - 3 PSK 27

Exrecise - 3 Example: A band pass data transmission scheme 33 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 Data rate =5000 bit/sec Receiver impulse response Threshold setting is 0 and 34 Dr. Uri Mahlab

=Probability of error = Pe :Solution Continue =Probability of error = Pe From the table of Gaussian probabilities ,we get Pe 0.0008 and Average error rate (rb) pe /sec = 4 bits/sec 35 Dr. Uri Mahlab

Exrecise - 4 Binary ASK 27

Binary ASK signaling schemes Exrecise - 4 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

Exrecise - 5 Power Spectrum 27

The power spectral density is given by Exrecise - 5 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

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

Exrecise - 6 Coherent ASK 27

Coherent ASK Exrecise - 6 41 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

43 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

Exrecise - 7 Noncoherent ASK

Noncoherent ASK The input to the receiver is Exrecise-7 44 Dr. Uri Mahlab

Non-coherent ASK Receiver Exrecise -8 Non-coherent ASK Receiver 27

Non-coherent ASK Receiver Exrecise-8 45 Dr. Uri Mahlab

: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

BINARY PSK SIGNALING SCHEMES Exrecise-9 BINARY PSK SIGNALING SCHEMES 27

BINARY PSK SIGNALING SCHEMES Exrecise-9 BINARY 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

Exrecises-10 Coherent PSK

Coherent PSK Exrecise-10 :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

DIFFERENTIALLY COHERENT PSK 27

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

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

Differential encoding & decoding 57 Dr. Uri Mahlab

BINARY FSK SIGNALING SCHEMES 27

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

Exrecise-11 Coherent FSK

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

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

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

Exrecise-12 Noncoherent FSK

Noncoherent FSK 63 Dr. Uri Mahlab

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

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

M-ARY SIGNALING SCHEMES 66 Dr. Uri Mahlab

M-ARY coherent PSK 66 Dr. Uri Mahlab

:The digital M-ary PSK waveform can be represented :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 Dr. Uri Mahlab

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

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

:The outputs of the correlators at time t=TS 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 4 8 16 32 0.5 0.333 0.25 0.2 0.34 dB 3.91 dB 8.52 dB 13.52 dB 76 Dr. Uri Mahlab Dr. Uri Mahlab

M-ary for four-phase Differential PSK

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

:The differential PSK waveform is :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 Exrecise 13 M-ary Wideband FSK Schemas 27

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

83 :N0 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

Pe1 = 1-Pc1 87 The probability that the receiver incorrectly * decoded the incoming signal S1(t) is Pe1 = 1-Pc1 The probability that the receiver makes * an error in decoding is Pe = Pe1 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

88 In the limiting case as M the probability of error Pe satisfies The maximum errorless rb 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

:Synchronization Methods For optimum demodulation of ASK ,FSK ,and PSK waveforms timing information is needed at the receiver There are three general methods used for synchronization in :digital nodulation schemes .Use of primary or secondary time standard .Utilization of a separate synchronization signal Extraction of clock information from the modulated waveform .itself , referred to as self - synchronization .1 .2 .3 89 Dr. Uri Mahlab

Open loop carrier recovery scheme (Extraction of local carrier for coherent demodulation of PSK signals) Open loop carrier recovery scheme Squaring circuit BPF Frequency divider Closed loop carrier recovery scheme Squaring circuit Loop Filter VCO Frequency doubler 90 Recovered carrier cos (wct) Dr. Uri Mahlab