1 Digital Communications69342 Part one-Pulse Modulation Dr. Allam Mousa An Najah National University Telecommunication Engineering Department Dig_Com_Pulse_Mod_1

2 1)Analog modulation ( AM, FM, …) 2) Pulse Modulation - Analog Pulse Modulation - Digital Pulse Modulation 3) Digital Modulation Dig_Com_Pulse_Mod_1

3 The transition from analog to digital communications. In continuous wave (CW) modulation, some parameters of a sinusoidal carrier wave is varied continuously in accordance with the message signal Analog Pulse Modulation: (Amplitude,duration, position) is varied. Information is transmitted basically in analog form, but the transmission takes place at discrete times. Digital Pulse Modulation The message signal is represented in a form that is discrete in both time and amplitude (coded pulses). Pulse Modulation Dig_Com_Pulse_Mod_1

Ideal sampled signal. Where δ(t-nTs) is delta function. Fourier Transformation Of gδ(t) G(t)  g(t) fs : Sampling Rate. Means : the process of uniformly sampling a continuous-time signal of finite energy results in a periodic spectrum with a period equal to the sampling rate. 4 1.Sampling Process Dig_Com_Pulse_Mod_1

Next: Gδ(f) = ∑ g(nTs) exp(-j2nπfTs) Discrete-Time Fourier Transform. Ts = 1/2ω ω is the largest frequency of G(f). fs = 2ω 5Dig_Com_Pulse_Mod_1

It can be shown that : Gδ(f) = fsG(f) + fs ∑G ( f – mfs) Let 1. G(f) = 0 for | f | >= 2. fs = 2ω Then If the sample values g(n/2ω) of g(t) are defined then G(f) is uniquely determined. If G(f) is defined then g(t) is determined. * Reconstructing g(t) from the sampled version g(n/2ω) Given G(f) then, G(t) = F-1[G(f)] 6Dig_Com_Pulse_Mod_1

The Sampling Theorem : for strictly band limited signals of finite energy. 1)A band limited signal of finite energy, with no frequency components higher than ω( Hz ) is completely described by specifying the values of the signal at instants of time separated by ( 1/ 2ω ) second. 2) A band limited signal of finite energy, with no frequency component higher than ω( Hz ), may be completely recovered from a knowledge of its samples taken at the rate of 2ω samples per second. Nyquist Rate fs = 2ω samples per second for a signal BW of ω Hz. Nyquist Interval 1/2ωTN = 1/fN = 1/2ω. 7Dig_Com_Pulse_Mod_1

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Aliasing: 9 If the signal is not strictly band limited. To prevent aliasing. 1. Prior to sampling, a low pass filter is used. 2. The filtered signal is sampled slightly higher than the Nyquist rate. fs > 2ω fs > fN Dig_Com_Pulse_Mod_1

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Sample And Hold 1. Instantaneous sampling of m(t) every Ts seconds. 2. Lengthening the duration of each sample to a constant value T ( to avoid an excessive band width channel ) "B.W is inversely proportional to pulse duration " BT = 1/T S(t) = ∑ m ( nTs ) h ( t – nTs ) Pulse Amplitude Modulation (PAM) 11Dig_Com_Pulse_Mod_1

Where Ts is sampling period. m(nTs) is the sample value of m(t) at time t = nTs. It can be shown that 12Dig_Com_Pulse_Mod_1

To reconstruct ( recover ) the original signal m(t) from s(t) H(f) = T Sinc(fT) exp ( -jπfT) So PAM introduces : - Amplitude distortion. -Delay of T/2. Distortion may be corrected by using an equalizer in cascade with LPF. 1/ |H(f)| = 1/ T Sinc (fT) = πf/ Sin(πfT) Magnitude response of the equalizer. If duty cycle T/Ts < 0.1, the amplitude distortion is less than 0.5% hence equalizer can be omitted. 13Dig_Com_Pulse_Mod_1

PAM : 1.Imposes requirement on the magnitude and phase responses of the channel. 2. Noise performance can not be better than baseband-signal Tx. 3. Suitable for TDM and producing some other forms of pulse modulation. 14Dig_Com_Pulse_Mod_1

Other forms of pulse modulation In pulse modulation we may use the increased bandwidth consumed by the pulses to improve the noise performance. Pulse Duration Modulation (PDM), also called (Pulse Width Modulation (PWM)): Samples of message signal are used to vary the duration of the individual pulses in the carrier 15 Modulating wave Pulse carrier PDM PPM Dig_Com_Pulse_Mod_1

Pulse Position Modulation (PPM) is more effective than (PDM) PPM is good if pulses are pure rectangles. No rising or falling time ( this requires infinite channel bandwidth ) So additive noise can effect only amplitude, but not position of rising or falling. Noise : figure of merit = ( SNR )0 / ( SNR )C. It's proportional to, so it may have some problems. [If average power of channel noise is small compared to the peak pulse power.] BT : transmission B.W. W : message B.W. provides corresponding quadratic increase (SNR)0 or figure of merit 16Dig_Com_Pulse_Mod_1

Bandwidth-Noise Trade-Off Noise performance : PPM system is the optimum form of analog pulse modulation. FM system is the optimum form of analog modulation. 1. Both PPM and FM have figure of merit proportional to. 2. Both exhibits a threshold effect as the SNR is reduced. Can we produce a trade-off better than a square-law? Yes : by using Digital Pulse Modulation  (Digital Modulation). 17Dig_Com_Pulse_Mod_1

Pulse Code Modulation : A message signal is represented in discrete form in both time and amplitude then noise effect at the receiver can be negligible. Now PCM uses an exponential law for the bandwidth noise trade- off. 18 Now PCM uses an exponential law for the bandwidth noise trade-off. Dig_Com_Pulse_Mod_1

Quantization Process: 19Dig_Com_Pulse_Mod_1

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Quantization noise : The difference between the input signal m & the quantized signal v. q =m-v → Q = M-V where m :amplitude of input signal let -m max < |m|< m min → where ∆ : step size, L : total # of representation levels quantization error : the probability density function : f Q = for this to be true the incoming signal does not overload the quantizer. 21Dig_Com_Pulse_Mod_1

If mean of quantization error = 0 Its variance IF R = # of bits per sample, L : total # of levels → →→ → 22Dig_Com_Pulse_Mod_1

Let P: average power of the message signal m(t), SNR at the output of the quantizer. Increasing R → increasing exponentially but also increasing the B.W R = 2 (binary code )provides more efficient method than FM or PPM. FM &PPM are limited by receiver noise. Binary code modulation is limited by quantization noise. 23Dig_Com_Pulse_Mod_1

→ →But increasing R requires a proportional increase in the channel (transmission) bandwidth → SO using binary (PCM) provides a more efficient method than either FM or PPM For the trade off of increased channel bandwidth for improved noise performance. FM & PPM are limited by receiver noise. Binary code modulation is limited by quantization noise. 24Dig_Com_Pulse_Mod_1

Example : Consider a full load sinusoidal signal with amplitude Am, using a uniform quantizer,find the (SNR)oq ? Average signal power, L =R =SNR SO changing R with the SNR : 25Dig_Com_Pulse_Mod_1

pulse code modulation (PCM) Is the most basic form of digital pulse modulation.. Sampling : 26 PCM Dig_Com_Pulse_Mod_1

Quantization : The sampled signal is quantized → discrete in both time &amplitude Nonuniform quantization passing the base band signal through a compressor then through a uniform quantizer. 27Dig_Com_Pulse_Mod_1

: Where Ѵ : normalized output : positive constant m: normalized input Uniform quantizer → A=1 or =0 At the receiver we use an “expander ” Compander compressor + expander ( = 255, =87.6) practical values 28 µ - law compressor Dig_Com_Pulse_Mod_1

“Encoding ” : After sampling & quantization, the signal is discrete in time & amplitude but not suitable to transmission over a telephone line or radio path. Encoding translates the discrete set of samples to a more appropriate form of signal (more robust to noise, interference & other channel Impairments ). Binary code is the best for noise immunity. Symbols may be binary, ternary,…….M -array R : # of bits → binary code → symbols. 29 Simple Code 0→ 000-4→ 100 1→ 001-5→ 101 2→ 010-6→ 110 3→ 011-7→ 111 Dig_Com_Pulse_Mod_1

Line coding : The choice of a particular pair of pulses to represent the symbols 1 &0 is called line coding. This depends on : 1.presence or absence of DC level 2.power spectral density 3.band width (spectral occupancy ) 4.Bit Error Rate (BER)performance 5.transparency (i.e,the property that any arbitrary symbol or bit pattern can be transmitted & received ) 6.easy of recovery (synchronization ) 30Dig_Com_Pulse_Mod_1

Split –phase (Manchester) 1 is represented by a positive pulse (of half duration ) of amplitude A followed by a negative pulse of half duration of amplitude –A, 0 the polarities are reversed. 31Dig_Com_Pulse_Mod_1

Commonly used line code symbols(0,1) and associated spectra 32Dig_Com_Pulse_Mod_1

Check if your solution of the previous example was correct. 33Dig_Com_Pulse_Mod_1

Differential encoding : A transition is used to designate symbol in the incoming steam of data & no transition is used to designate symbol 1. Whenever you encounter a zero in the incoming signal more transition in the encoded signal. The decoder compares the polarity of adjacent binary symbols to establish whether or not a transition has occurred (a reference bit is required ) 34Dig_Com_Pulse_Mod_1

PCM transmission path : 35Dig_Com_Pulse_Mod_1

From fig. the differentially encoded signal may be inverted without affecting its interpretation The original binary information is recovered simply by comparing the polarity of adjacent binary symbols to establish wither as atransition has occurred. Regeneration enables a PCM system to control the effects of distortion and noise produced by transmitting a PCM signal through a channel. 36Dig_Com_Pulse_Mod_1

Equalizer : shaper the received pulse to compensate for the effects of amplitude & phase distortions (due to channel ) Timing circuit : provides a pulse train for sampling the equalized pulses at max SNR time. Decision device : the threshold, comparing …………. No accumulation of distortion Regeneration → reshape & clean up received pulses. In practice : bit error is introduced if we have an unavoidable channel noise & interference jitter is introduced if the spacing between received pulses & deviates from its assigned value. 37Dig_Com_Pulse_Mod_1

Decoding : Regrouping the pulses into code words ( quantized PAM signal ) 0101 → decoder →amplitude → decoder →amplitude 6 Filtering : To recover the message signal wave by passing the decoder output through a low –pass reconstruction filter with cut off frequency = message B.W (W) 38Dig_Com_Pulse_Mod_1

Multiplexing : In PCM,multiplexing different message sources by Time Division Multiplexing (TDM) Message sources may be dropped or reinserted easily in TDM. As the number of message is increased, the time interval that is allocated to each source has to be reduced. (all messages must be accommodated into a time = Ts). This reduces the duration of each code word →reducing pulse duration → pulses become more difficult to generate → B.W increases → interfering with other signals. Synchronization : The timing operation at the receiver, follows closely the corresponding operations at the transmitter(except for time lost in transmission & regeneration) To synchronize Tx &Rx we can send a pulse at the end of a frame (transmitted every other frame only ). 39Dig_Com_Pulse_Mod_1

noise consideration in PCM systems : Major noise sources : channel noise, always present quantization noise, introduced in the transmitter (signal dependent ) the noise cause bit error (0→1,1→0) Define : Average probability of symbol error as the probability that the reconstructed symbol at the receiver output differs from the transmitted binary symbol, on the average (Bit Error Rate BER ) Usually channel nose is assumed additive, white,& Gaussion Quantization noise is under designer's control. 40Dig_Com_Pulse_Mod_1

Error threshold : BER depends on Eb/No,where Eb/No is the ratio of the transmitted signal energy per bit,Eb, to the noise spectral density,No. Influence of Eb/No on the probability of error : For abit rate of 105 b/s, one error every probability of error PeEb/No (db) 10-3 second second second ………………………………………→ minutes day months Dig_Com_Pulse_Mod_1

The threshold is about Eb/No = 11.0 dB If Eb/No > 11.0 dB, channel noise has virtually no effect on the receiver performance ( goal of PCM ) CASE : 11 dB error threshold in PCM,NRZ signaling 60 dB error threshold in amplitude modulation, high quality, speech → PCM requires much less power but average noise power is increased due to the increase in bandwidth by ???? PCM is robust to channel noise & interference 42Dig_Com_Pulse_Mod_1