EEE 309 Communication Theory Semester: January 2017 Dr. Md. Farhad Hossain Associate Professor Department of EEE, BUET Email: mfarhadhossain@eee.buet.ac.bd Office: ECE 331, ECE Building
Types of Modulation 1. Analog/Continuous Wave Modulation: Analog baseband signal using analog carrier (bandpass channel) A. Amplitude Modulation (AM) B. Angle Modulation i) Frequency Modulation (FM) ii) Phase Modulation (PM) 2. Digital Modulation: Digital bit stream using analog carrier (bandpass channel) A. Amplitude Shift Keying (ASK) B. Frequency Shift Keying (FSK) C. Phase Shift Keying (PSK) 3. Pulse Modulation: Analog narrowband signal using pulse (wideband baseband channel) A. Pulse Amplitude Modulation (PAM) B. Pulse Time Modulation (PTM) i) Pulse Width Modulation (PWM) / Pulse Duration Modulation (PDM) ii) Pulse Position Modulation (PPM) C. Pulse Code Modulation (PCM) digital pulse modulation D. Delta Modulation (DM) 2
Part 05 Pulse Code Modulation (PCM) 3
Pulse Code Modulation (PCM) In PCM, a message signal is represented by a sequence of coded pulses, which is accomplished by representing the signal in discrete form in both time and amplitude Transmitter Three basic operations in a PCM Transmitter: Sampling Quantization Encoding Transmission Path Receiver 4
Sampling Sampling is an operation that is basic to digital signal processing and digital communications Through the use of sampling process, an analog signal is converted into a corresponding sequence of samples that are usually spaced uniformly in time Two types of practical sampling: Natural Sampling Flat-top sampling 5
Natural Sampling (1) Message Sampling Signal T s f f f nf s n s Sampled Signal 6
Natural Sampling (2) Frequency Domain: or, 7
Natural Sampling (3) f s > 2W: f s = 2W: f s < 2W: Aliasing 8
Sampling Theorem Sampling theorem is a fundamental bridge between continuous signals (analog domain) and discrete signals (digital domain) It only applies to a class of mathematical functions whose Fourier transforms are zero outside of a finite region of frequencies Nyquist Sampling Theorem / Nyquist-Shanon Sampling Theorem: A signal whose bandwidth is limited to W Hz can be reconstructed exactly (without any error) from its samples uniformly taken at a rate f s 2W Hz f s = Sampling frequency f s = 2W: Nyquist frequency / Nyquist rate / Minimum sampling frequency 9
Antialiasing Filter All practical signals are time-limited, i.e., non band-limited => Aliasing inevitable To limit aliasing, use antialiasing filter (LPF) before sampling Original Signal Antialiasing filter Sample Reconstruction Filter Reconstructed Signal 10
Reconstruction Filter f s = 2W: Ideal LPF characteristic: 1/2W 1/2W (interpolation filter / interpolation function) T s = 1/2W (interpolation formula) 11
Quantization (1) It is the process of transforming the sample amplitude m(nt S ) of a baseband signal at time t = nt S into a discrete amplitude v(nt S ) taken from a finite set of possible levels 12
Quantization (2) Quantizer characteristic: k th interval: Here, k = 1, 2, 3,, L L = Number of representation levels (Number of intervals) m k : Decision levels / Decision thresholds v k : Representation levels / Reconstruction levels / Quantization Levels Δ= v k +1 v k = m k +1 m k : Step-size / quantum Quantizer output equals to v k if the input signal sample m belongs to the interval I k (rounding) v v if m I (Rounding) k I k v v k if v k m v k 1 (Truncation) 13
Quantization(3): Two types Mid tread quantization Mid rise quantization Representation/ Reconstruction/ Quantization levels Mid tread quantizer: Mid rise quantizer: Reconstruction value is exactly zero Decision threshold value is exactly zero Signal Range (Dynamic range) and Quantizer Range: Could be same or different 14
Quantization(4): Example For the following sequence {1.2, -0.2, -0.5, 0.4, 0.89,1.3} quantize it using a uniform quantizer of rounding type and write the quantized sequence. Quantizer range is (-1.5,1.5) with 4 levels. Solution: Yellow dots indicate the decision thresholds (boundaries between separate quantization intervals). Red dots indicate the reconstruction levels (middle of each interval). Thus, 1.2 fall between 0.75 and 1.5, and hence is quantized to 1.125. Quantized sequence: {1.125, -0.375, -0.375, 0.375, 1.125, 1.125} 15
Quantization(5): Two types Uniform quantization Non uniform quantization 16
Quantization Error for Uniform Quantization (1) Quantization error (noise) q = m v => Q = M V Q is a RV variable of zero mean in the range [ Δ/2, Δ/2] If Δ is sufficiently small, Q can be assumed a uniform RV with zero mean Quantization noise power f Q (q) 1/Δ 2 Q / 2 2 2 q f Q qdq 12 / 2 Δ/2 0 Δ/2 q Signal-to-nose-ratio (SNR): SNR P 2 Q 12P P 2 P = Average power of m(t) 17
Quantization Error for Uniform Quantization (2) Suppose m(t) of continuous amplitude in the range [ m max, m max ]: 2mmax 2mmax R L 2 R = Number of bits for presenting each level (bits/sample) SNR 3P 2 m 2 max 2R SNR db 3P 6.02R 10log m 2 max Each additional bit increases the SNR by 6.02 db and a corresponding increase in required channel BW Special case: m(t) is a sinusoidal signal with amplitude equal to m max SNR 3 2 2 2R 6R 1. 8 SNR db 18
Non-Uniform Quantization SNR of weak signals is much lower than that of strong signal Instantaneous SNR is also lower for the smaller amplitudes compared to that of the larger amplitudes 19
Non-Uniform Quantization Step size increases as the separation from the origin of the input output amplitude characteristic is increased First Compression and then uniform quantization Achieve more even SNR over the dynamic range using fewer bits (e.g., 8 bits instead of 13/14 bits) Receiver side: Expansion required Compression + Expansion = Companding Tx Rx Original Signal Compression Uniform Quantization Reconstruction Expansion Original Signal 20
What is Compression? The compression here occurs in the amplitude values Compression in amplitudes means that the amplitudes of the compressed signal are more closely spaced in comparison to the original signal To do so, the compressor boosts the small amplitudes by a large amount. However, the large amplitude values receive very small gain and the maximum value remains the same Compressor Input Compressor Output 21
Non-Uniform Quantization μ Law: Used in North America, Japan (μ = 255 is mostly used) Compression characteristics (first quadrant shown) More uniform SNR is achieved over a larger dynamic range 22
Non-Uniform Quantization A Law: Used in Europe and many other countries A = 87.6 is mostly used and comparable to μ = 255 23
Encoding Each quantized samples is encoded into a code word Each element in a code word is called code element Binary code: Each code element is either of two distinct values, customarily denoted as 0 and 1 Binary symbol withstands a relatively high level of noise and also easy to regenerate Each binary code word consists of R bits and hence, this code can represent 2 R distinct numbers (i.e., at best R bit quantizer can be used) 24
Pulse Code Modulation (PCM) In PCM, a message signal is represented by a sequence of coded pulses, which is accomplished by representing the signal in discrete form in both time and amplitude Transmitter Three basic operations in a PCM Transmitter: Sampling Quantization Encoding Transmission Path Receiver 25
Differential PCM (DPCM) When a signal is sampled at a rate slightly higher than the Nyquist rate, there exists a high degree correlation between adjacent samples, i.e., in an average sense, the signal does not change rapidly from one sample to the next When these highly correlated samples are encoded as in a standard PCM system, the resulting encoded signal contains redundant information implying that symbols that are not absolutely essential to the transmission of information are generated DPCM removes this redundancy before encoding by taking the difference between the actual sample m(nt S ) and its predicted value mˆ nt S The quantized version of the prediction error e(nt S ) are encoded instead of encoding the samples of the original signal This will result in much smaller quantization intervals leading to less quantization noise and much higher SNR Transmitter e Prediction error nt mnt mˆ nt S S S 26
Predictor for DPCM: Liner predictor of order p: Transversal filter (tapped-delay-line filter) used as a linear predictor p nt w m n k mˆ nt S k 1 k q T S Past p samples 27
Differential PCM (DPCM) Transmitter e q nt S m ' nt S mˆ nt ' Receiver S m nt ˆ nt nt S m S eq S Reconstruction error ' m nt m nt ent e nt qnt S = Quantization error S S q S S 28
Delta Modulation (DM) (1) DM encodes the difference between the current sample and the previous sample using just one bit Correlation between samples are increased by oversampling (i.e., at a rate much higher, typically 4 times higher than the Nyquist rate) DM involves the generation of the staircase approximation of the oversampled version of message The difference between the input and the approximation is quantized into only two levels: 1 bit version of DPCM (i.e., 2 level quantization) requiring less bandwidth than that of DPCM and PCM 29
Delta Modulation (DM) (2) Transmitter Receiver Digital equivalent of integration 30
Predictor for DPCM and DM DM DPCM Note: (1) DPCM uses a higher order filter. (2) DM uses a 1 st order (p=1) predictor with w 1 = 1. Thus, the predicted output is the previous sample. 31
Delta Modulation (DM) (3) Two types of quantization error: (1) Slope overload distortion/noise (2) Granular noise m q (t) e q (nt S ) Comments: (1) For avoiding slope overload distortion: larger Δ is desired (2) For avoiding granular noise: smaller Δ is desired An optimal step size (Δ) has to be chosen for minimum overall noise Example: Avoiding slope overload mt A t m t max m Am f cos s A m m m T A m f s f s max Voice r 2 800 r T s max m 32
Line Coding (1) PCM, DPCM and DM are different strategies for source encoding, which converts an analog signal into digital form Once a binary sequence of 1s and 0s is produced, the sequence is transformed into electrical pulses or waveforms for transmission over a channel and this is known as line coding Multi-level line coding is possible Or NRZ-L Various line coding (binary) methods: Or RZ-AMI (0 means transition) (f) Split-phase or Manchester 33
Line Coding (2) Polar NRZ / Book: Digital Communications: Fundamentals and Applications - Bernard Sklar Bipolar NRZ Applications: Polar NRZ / NRZ-L: Digital logic circuits NRZ-M/NRZ-S: Magnetic tap recording RZ line codes: Base band transmission and magnetic recording (e.g., Bipolar RZ / RZ-AMI is used for telephone system) Manchester Coding: Magnetic recording, optical communications and satellite telemetry 34
Line Coding (3) 35
Line Coding (4) Desired properties (i.e., design criteria) for line coding: Transmission bandwidth: should be as small as possible Noise immunity: should be immune to noise Power efficiency: for a given bandwidth and given error probability, transmission power requirement should be as small as possible Error detection and correction capability: should be possible to detect and correct errors Favorable power spectral density (PSD): should have zero PSD at zero (i.e., DC) frequency, otherwise the ac coupling and the transformers used in communication systems would block the DC component Adequate timing information / self-clocking: should carry the timing or clock information which can be used for self-synchronization Transparency: should be possible to transmit a digital signal correctly regardless of the patterns of 1 s and 0 s (by preventing long string of 0s and 1s) 36