Pulse Modulation

1. Pulse Modulation

2. Introduction • In Continuous Modulation C.M. a parameter in the sinusoidalsignal is proportional to m(t) • In Pulse Modulation P.M. a parameter in the pulse train is proportional to m(t) • In analog P.M. the parameter (amplitude, position, duration) is varied in a continuous manner • In digital P.M. the values are discrete values • P.M. is a transition between analog modulation and digital modulation

3. Kinds of Pulse Modulation • PAM Pulse Amplitude Modulation • PDM Pulse Duration Modulation (or) PWM Pulse Width Modulation • PPM Pulse Position Modulation • PCM Pulse Code Modulation

5. Natural Sampling • The sampled signal consists of a sequence of pulses of varying amplitude whose tops are not flat but follow the shape of the waveform of the signal m(t).

6. Flat-Top Sampling • The sampled signal consists of a sequence of pulses of flat tops amplitude. • It will make distortion for recovered signal, but the distortion will not be noticeable when the number of samples are large,

7. Pulse Amplitude Modulation(PAM) • The amplitudes of regularly spaced pulses are varied in proportion to the corresponding sampling values of a continuous message • Similar to natural sampling: the message signal is multiplied by a train of rectangularpulses. The top of each modulated rectangle is maintained notflat

9. Two operations evaluated in the generation of PAM: - Instantaneous sampling every Ts - Lengthening the duration of the sample to some constant value T. • To reconstruct the signal, we need an equalizer (since the sample and hold filter used at the transmitter alter the shape of the signal) plus the reconstruction filter

10. Reconstruction filter Equalizer PAM signal • The reconstruction filter is an ideal L.P.F having a cut off frequency equals to the signal bandwidth. • The equalizer frequency response: Heq(f) = 1/ | H0(f)| = 1 / [ T sinc (fT ) = π f / [ sin (π f T) ]

11. Quantizer g(.) Continuous sample m Discrete sample v Quantization Process • Quantization is to approximate each sample to the nearest level. • Amplitude quantization: The process of transforming the sample amplitude m(nTs) of a message signal m(t) at time t = nTs into a discrete amplitude v(nTs)taken from a finite set of possible amplitudes

13. Define: • LTotal number of amplitude levels used in the quantizer • mk discrete amplitudes k = 1,2 …L (decision levels, thresholds ) • vk representation (or reconstruction) levels • Ik quantizer interval vk-2 vk-1 vk vk+1 mk-1 mk+2 mk mk+1 Ik

14. Step size (quantum): spacing between two representation levels • The quantizer O/P v = vk if the input signal m belongs to the interval Ik

15. Uniform (linear) Quantization The representation levels are uniformly spaced • Mid-tread: Origin in the middle of staircase • Mid-rise: Origin in the middle of rising part of staircase Mid-treadMid-rise

16. Quantization Noise for Uniform Quantization • The use of quantization introduces an error between the I/P signal m(t) and the signal at quantizer output mq(t). • This error is known as: quantizationerror or (e), where: e = m(t) – mq(t)

17. Quantization Noise for Uniform Quantization • If m(t) is in the range ( - mmax, mmax ) Then the Stepsize of the quantizer : S = 2 mmax / L • Assume that the quantization error is uniformly distributed within each quantization range, then its p.d.f f(m) will be: f1(m)+f2(m)+ f3(m)+….+ fL(m) = 1 / S f(m) f1(m) f2(m) f3(m) fL(m) f1(m) f4m) m s

18. σQ2=Quantization noise = mean square value of the error = mmax2/ [ 3 L2 ] [S=2mmax/L] • If ‘n’ is the number of bits per sample L = 2n S = 2mmax / 2n σQ2 = mmax2 2-2n/ 3

19. Let P = average power of the message signal m(t) • Assuming m(t) is uniformly distributed from –mmax to mmax • Output signal to noise ratio = SNRo/p = P /σQ2 = 3 P 22n/ mmax2 = 3 K (2)2n • Where: K = P / mmax2 = 1/3 • [SNRo/p]= 10 log [(2)2n] = 6n (dB)

20. Non-Uniform quantization(Companding) • Why? • Low amplitudes happens more frequently than larger ones • Since the mean square error is proportional to the step size, we need to decrease the step size for lower values than larger values. • This is accomplished by using a compressor at the transmitter and an expander at the receiver • The combination of a compressor and an expander is called compander

21. Non-Uniform quantization(Companding) • Two types of compressors may be used: • A law and μ law refer to the parameter which appear in the equation of compression and expansion. • -Law A-Law

25. Non-Uniform quantization(Companding) • Dynamic Range: is the difference in dB between max. signal level, and min. signal level having accepted output SNR . • To improve dynamic range Compandingis used to keep SNR high for low signal level as well as for high signal level

26. Non-Uniform quantization(Companding) Output SNR (dB) Companded 48 10 dB 30 Si(dB) normalized input signal power in dB -48 -18 0 Un-Companded

27. Non-Uniform quantization(Companding) • Example: • For acceptable voice transmission the received signal have a ratio SNR > 30 dB. • The compandedsystem has dynamic range of input signal=48 dB. • The un-compandedsystem has dynamic range of input signal=18 dB for the same condition of SNR>30 dB • The penalty paid is at max. amplitude of input signal, SNR is less 10 dB after companding.