Communication Systems: Digital Representation of Analog Signals (PDF)
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BITS Pilani
Prasant Kumar Pattnaik
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These lecture notes cover the digital representation of analog signals, exploring concepts like sampling, quantization, encoding, and transmission methods. The document provides a detailed overview, suitable for an undergraduate-level course in electrical engineering.
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Communication Systems Prof. Prasant Kumar Pattnaik Associate Professor Department of Electrical and Electronic Engineering BITS Pilani Hyderabad Campus ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Topic # 6: Digital Repres...
Communication Systems Prof. Prasant Kumar Pattnaik Associate Professor Department of Electrical and Electronic Engineering BITS Pilani Hyderabad Campus ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Topic # 6: Digital Representation of Analog Signals T1. B.P. Lathi, Modern Digital and Analog Communication Systems, 3rd Edition, Oxford University Press, 1998: OR 4th Edition 2010 Chapter 6 T2. Simon Haykin & Michael Moher: Communication Systems; John Wiely, 4th Edition OR 5th Edition, 2010, 5/e. : Chapter 7 2 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Why Digital Communications? 1. Can withstand channel noise and distortions better than analog systems. 2. Amenable to the use regenerative repeaters for long distance communications 3. Implementations are flexible and one can use m - processors and VLSI systems 4. Error correction coding, multiplexing, storage and reproduction of data and efficiency in SNR and BW exchange, etc. are superior to analog systems 3 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Digital Transmission of Analog Signals An analog signal can take any amplitude value over a continuous range. It means that it can take on an infinite number of values. To transform an analog waveform into a form that is compatible with a digital communication system, the following steps are taken: 1. Sampling 2. Quantization 3. Encoding 4. Baseband OR Passband transmission 4 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Digital Transmission of Analog Signals 5 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sampling Theorem A signal g(t) whose spectrum is band limited to B Hz can be reconstructed from its samples taken uniformly at a rate R samples per second (spaced uniformly at an interval Ts < 1/2B) such that In other words, the sampling frequency, fs ≥ 2B. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sampling Theorem Sampling at fs Hz means taking fs samples per second. This can be accomplished by multiplying g(t) by an impulse train as shown in figure: The resulting sampled signal is shown in fig(d) The relationship between sampled signal and original signal is: Therefore, ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sampling Theorem By using the frequency shifting property, the Fourier transform of g(t) is: For perfect recovery of g(t) from , there should not be any overlapping among The replicas of in ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sampling Theorem It requires, fs ≥ 2B. Here G(f) does not contain any impulse at the highest frequency, hence the sufficient condition is fs = 2B. But if G(f) contains an impulse at the highest frequency, then the condition should be fs ≥ 2B. For example,g(t) = sin2πBt contains impulses at ±B Hz, so if we take fs = 2B, The impulses will cancell each other and spectra will be zero. Hence, fs ≥ 2B. The minimum sampling frequency for signal reconstruction is called Nyquist frequency and corresponding sampling interval is called Nyquist interval. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sampling Process FT Pair T s (t nT ) n s ωs-B ≥B 11 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Signal Reconstruction The process of reconstructing the continuous signal g(t) from its samples is known as Interpolation. The signal g(t) can be reconstructed from its sampled version by passing it through an Ideal low pass filter as shown: The ideal low pass filter should have a bandwidth B Hz and a gain of Ts. The frequency response of such a filter is: ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Ideal Reconstruction Again, the frequency response of ideal low pass filter for signal reconstruction Is: Hence, the impulse response is: Here, h(t) = 0 at the Nyquist instants except at t = 0. So, when sampled signal is applied as the input, output will be g(t). ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Ideal Reconstruction Since the filter output is the convolution of sampled signal with h(t), each sample (impulse) in generates a sinc pulse of height equal to strength of the sample as shown in figure: The kth sample of sampled signal is the impulse , so the filter output for this impulse is: ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Ideal Reconstruction So, filter output corresponding to all the impulses must be equal to g(t). The above equation is the interpolation formula which gives the value of g(t) as The weighted sum of all the sample values. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Recovery, Nyquist Rate & Nyquist Interval Recovery of the un-sampled signal from the samples is possible by passing through the ideal low pass filter (reconstruction filter) s m m Provided : fs > 2B : Nyquist rate Ts < 1/ 2B : Nyquist Interval Sampling Theorem 16 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Recovery: As Interpolation from Samples Pass through LPF Output of the LPF With Nyquist Sampling rate Output of the LPF is 18 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Recovery: As Interpolation from Samples yields values of g(t) between samples as a weighted sum of all the sample values. Interpolation 19 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Recovery: As Interpolation from Samples yields values of g(t) between samples as a weighted sum of all the sample values. Interpolation Formula: 20 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Simpler Interpolation from Samples Zero Order Hold filter First order hold filter is using liner approximation between samples. 21 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical signal Reconstruction 22 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical signal Reconstruction g~ (t ) g nTs pt nTs n g~ (t ) p (t ) g nTs t nTs p(t ) g t n ~ Gf Pf 1 Gf nf s T s n ~ G f E f G f E f P f 1 G f nf s T s n E f P f 0 E f P f Ts f fs B f B p f T p sinc fT p e jfT p t 0.5T p pt E f Ts / P f f B T E f Ts / T p p f B t nTs 0.5T p g~ (t ) g nTs n Tp 23 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical Issues 1. Realizability of Reconstruction Filters: By sampling the signal g(t) at Nyquist rate, the spectrum of sampled signal consists of replicas of G(f) without any gaps between them. To reconstruct g(t), an ideal low pass filter (with sharp cut-off) is required. Such a filter is unrealizable in practice. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical Issues The solution is to sample the signal g(t) at a rate higher than the Nyquist rate and use the practical low pass filter with gradual cut-off characteristics as shown: But in this case also, filter gain is required to be zero beyond the first cycle of G(f). According to Paley-Wiener Criterion, it is impossible to realize even this filter. This shows that it is impossible, in practice, to recover the band limited signal g(t) exactly from its samples. However, as sampling rate increases, the recovered signal approaches the desired signal more closely. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical Issues 2. Aliasing: In sampling theorem, we assumed the signal g(t) to be band limited to B Hz. But, all practical signals are time-limited and a time-limited signal can not be band-limited. Clearly, the signal g(t) has an infinite bandwidth and we can not avoid the overlapping of the spectra of G(f) in the sampled signal. So, the reconstructed signal will be different from the original signal g(t) even if we use a low pass filter that closely approximates an ideal filter. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical Issues The reconstructed signal will have two problems: (i) Loss of tail of G(f) beyond |f| > fs/2 (ii) Reappearance of this tail as folded or inverted back into the spectrum. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical Issues This tail inversion is called spectrum inversion or aliasing. Such aliasing destroy the integrity of the frequency components below folding frequency (fs/2). ( Recovered Signal) ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Anti-aliasing Filter This is a low pass filter with sharp cut-off at the folding frequency fs/2. The signal g(t) to be sampled is first passed through the anti-aliasing filter to suppress the frerquency components beyond fs/2. In this way, g(t) is now a signal that is band-limited to fs/2 Hz. By using this technique, we will loose the components beyond fs/2 Hz but at the same time prevent the lower frequency components from being damaged in the reconstructed signal. The reconstructed signal spectrum is Gaa(f) = G(f) for |f| < fs/2. The anti-aliasing filter also helps to reduce noise because it suppresses the noise components above the frequency fs/2. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Antialiasing Filter ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Practical issues with Sampling. Realizability of Reconstruction filter 31 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Natural Sampling / Sample & Hold FT Pair 32 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Under sampling & Aliasing fm Correctly A realizable LPF dictates the Need for Fs > 2B Sampled Spectrum Fs = 2fm Under Sampled A time limited signal will have spectrum Spectrum extending to infinity Since one can not sample at infinite frequency, under sampling prevails Fs < 2fm What is the Effect of Under sampling ? 33 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Under Sampling & Aliasing Resultant Message Spectrum Spectrum Sampled Spectrum Fs = 2fm - fm fm Recovered Aliased message Spectrum Spectrum To avoid this, Anti Aliasing Filters are needed 34 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION How to Avoid Aliasing – Before Sampling Message Spectrum LPF - fm fm Fs Fm = Fs /2 Anti Aliasing Filter – Before Sampling 35 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION How to Avoid Aliasing – After Sampling Message Spectrum Sampled Spectrum - f’m - fm fm f’m F’s = 2f’m Aliased Anti Aliasing Filter – After Sampling Spectrum 36 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sampled Signals & Pulse Modulations PDM / PWM: Sample values are used to The signal g(t) is sampled modify width of the pulses of a periodic pulse train. PAM: Sample values are used to modify PPM: Sample values are used to modify amplitudes of pulses of a periodic pulse the positions of pulses of a periodic train. pulse train. Bandwidth required for transmission is inversely proportion to the least width of the pulse used !! 37 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION TDM with PAM Several PAM signals can be Time Multiplexed (TDM) 38 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION TDM with PAM 39 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION TDM with PAM Figure shows a PAM-TDM representation 1ms ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Digital Transmission of Analog Signals An analog signal can take any amplitude value over a continuous range. It means that it can take on an infinite number of values. To transform an analog waveform into a form that is compatible with a digital communication system, the following steps are taken: 1. Sampling 2. Quantization 3. Encoding 4. Baseband transmission 41ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Quantization Map the samples of a continuous amplitude waveform to a finite set of amplitudes. Output: finite set of amplitudes Dv = 2 mp / L. Input: continuous amplitudes 43 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Quantization Number of quantization levels L = 8 44 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Quantization L is the number of quantization levels Instantaneous Quantization error is Assuming that the error is equally likely to lie anywhere in the range (-Dv/2, Since Dv/2), the mean square of quantization error is =(1.8+6n)dB 45 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION 9/25/2024 46 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION How many Quantization Levels ? If the magnitude of the quantization error | e| is specified as a fraction p of the peak to peak analog voltage 2 mp as | e| ≤ p (2 mp) | e|max ≤ Dv/2 p (2 mp) ≤ L ≥ 1 / 2p 47 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Quantization 48 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Is Uniform Quantization Always Good ? Using equal step sizes (uniform quantizer) gives Signal to quantization noise ratio is low SNR for weak signals and high SNR for proportional to the signal strength. strong signals. Since, Signal to quantization noise ratio is inversely proportional to the square of the step size, Can we reduce the step size for weaker portions of the signal ? SNR is proportional to 1/[(ΔV)2/12] In speech, weak signals are more frequent than strong ones. Statistics of Speech amplitudes Non Uniform Quantization ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Non-uniform Quantization Adjusting the step size of the Transmitter quantizer by taking into account y = C(x) the speech statistics improves the SNR for the input range. x(t) y(t) It is achieved by uniformly quantizing the “compressed” x signal. Compress Uniform Qauntizer Receiver xˆ(t) xˆ (t ) Compression + expansion => x̂ companding Expand ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Companding - Compression + Expansion m - law compression A - law compression 51 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Sources of errors while Quantizing 1. Quantization Noise: 2. Quantizer Saturation: The signal amplitude crosses the maximum handling level of the quantizer Those amplitudes of the signals are saturated to either maximum or minimum levels and hence distortion. 3. Timing Jitter: Sampling instants are not stable in time. Random jitter Causes low level additional spectral components and hence noise. Periodic jitter appears as an extra sinusoid at the receiver. 52 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Encoding The quantized sample values can be transmitted using PAM This calls for multi-level voltage values to be sent over line Sending a minimum number of voltage levels over line has advantages. In Binary PCM, each quantized sample is digitally encoded into an n bit code word. If L in the number of quantization levels, number of bits per sample: 53 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Binary PCM Encoding Number of quantization levels L = 8 : number of PCM bits : n = log2L = 3 54 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Transmission BW and Number of bits/sample in PCM To send B Hz signal, over a noise free channel, without any errors, one needs a channel bandwidth of B Hz. Nyquist Sampling rate is 2B samples /sec. => A channel with a BW of B Hz can support data rate of 2B samples / sec In Binary PCM, each sample is encoded into n bits. Thus, a signal, band limited to B Hz, generates 2nB bits / second. ( 2nB information bits / second) Hence, the required minimum bandwidth for transmission, BT = nB Hz. => Given nB Hz bandwidth, we can transmit 2nB bits /second. => 2 bits/sec/Hz is the BW efficiency. 55 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Transmission BW Relations 1. Band limited signal, limited to B Hz 3. Each sample is encoded into n bits, hence, 2nB information samples are generated per second. Maximum Inter bit duration Tb = Ts/n = 1/nFs = 1/2nB sec Ts = 1/Fs = 1/2B 4. Hence, 2nB samples /sec information rate requires nB Hz bandwidth. This is called Nyquist bandwidth. Minimum bit rate: Rb = 1/Tb = 2nB bits/sec 2. For error free transmission, with no noise present, 2B samples /sec Minimum BW ( Nyquist BW) : Rb /2 Hz. information rate requires B Hz band width 56 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Transmission BW & SNR ( quantization) Signal to quantization noise ratio in uncompressed case is Signal to quantization noise ratio in m-law compressed case is The number of levels L is related to the number of bits in PCM word as L = 2n uncompressed m-law The transmission bandwidth BT = nB Signal to quantization noise ratio can be improved, exponentially, by linear increase in transmission BW. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Transmission BW & SNR ( quantization) An increase in the code word size by 1 bit, the SNR (quantization) increases by 6 dB. An increase in the code word size by 1 bit, the BW increase is (1/n) Example: For an increase from 8 to 9 bits, SNR increases by 6dB and demands an increase BW by 12.5 % ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Encoding of Voice / Speech They are divided into two main categories: PCM is a type of Voice Encoder 1. Waveform Coders The analog voice is converted to n-bit PCM and transmitted 2. Model based Coders. However, the PCM is not efficient as it requires bitrate Rb = 2nB bits/sec (nB Hz 1. Waveform Coders: Purely based on the Transmission bandwidth). waveforms as they are available. They do not consider how the waveform / signal was actually generated. Ex: For 8 KHz sampled signal with 8 bits per sample, the transmission bit rate is 2. Model Based Coders: The signal / Rb = 64 kbits/sec. (64 kbps) waveform generation process is understood accordingly the coding is done. There exist other methods that result in to less bit rates. They exploit the inherent Both of the above are also considered characteristics of the underlying signal. as Compression Techniques. 59 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: Differential PCM (DPCM) Let m(k) be the kth sample of a signal m(t). Instead of transmitting m(k), can we transmit d(k) = m(k)-m(k-1) ? Since, d(k) is usually much smaller than m(k), quantization error in d(k) is less. Also, d(k) may require fewer number of bits to encode: => reduced the Bit rate Rb At the receiver, from the knowledge of m(k-1) and d(k), one can reconstruct m(k). The prediction error is It can be shown using Taylor series expansion and other theories that, for a If we transmit the prediction error d(k) , signal that has certain amount of inherent the receiver can add the d(k) to the correlation, the kth sample value can be locally generated estimate of m(k), to get estimated using an Nth order predictor as: back m(k) 60 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: Prediction & DPCM At Transmitter At Receiver 61 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: Prediction & DPCM At Transmitter With DPCM, the quantization step size is reduced for a given number of levels L. This implies that the quantization Noise is reduced and the corresponding SQNR will improve For the same number of bits/ sample, the SQNR improves (over PCM) by about 5.6 dB for a 2 step predictor. At Receiver Alternately, for the same SQNR, the required quantization levels and subsequent number of PCM bits will decrease. For the same SQNR , we require 3-4 bits/ sample less than PCM 62 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: ADPCM In DPCM, The quantization step is fixed. ADPCM Thus, compared to DPCM, ADPCM can further compress the bit rates. International Telecommunication Union, (ITU) specifies the standards and adapted However, the prediction error could be ADPCM under G-726 Standard. small or large, depending on the signal and the predictor accuracy. G.726, for a 8 kHz sampled voice, has an 8th order predictor and can support bit rates as: 16, 24, 32 and 40 kbps. Suppose, the quantization step is made adaptive, depending on the prediction error, then number of bits can be This implies, 2, 3, 4 and 5 bits / sample. reduced. Standard PCM has 8 bits/ sample. 63 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: Delta Modulation For a baseband signal, oversampled, by say, k times, the correlation between adjacent samples is high. => prediction error is low. Can afford to use a 1 step predictor. For the reduced prediction error, it is sufficient to code using a single bit PCM. At Transmitter At Receiver 64 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: Delta Modulation Slope Overload Issue At Transmitter At Receiver 65 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Remedies for Slope overload in DM Slope overload occurs when q (t) can not follow m(t). For Voice signals, with BW = 4KHz, it is observed that Single integrated overload characteristics During sampling interval, Ts, q (t) is capable of changing by one step size, s. Voice spectrum Maximum slope of m(t), that q (t) can follow is s / Ts = s fs No Overload condition is double integrated overload characteristics For a single tone For no Overload Use single Integration up to 2000 Hz and A max without overload is double Integration afterwards 66 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Waveform Encoder: Adaptive DM In delta modulation, the variations in m(t), smaller than step size are lost. – Threshold effect On the other hand, if variations in m(t) are too fast, q (t) can not follow the variations - Overload effect Solution is to go for adaptive step size of s according to the level of the signal derivative. 67 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION Model Based Encoding Waveform Coders do not consider how the At the transmitter, from the knowledge waveform / signal was actually generated. of the segment of speech and the Hence, the level of compression is limited. sequence of impulses, the system parameters are identified. Even with minimum Nyquist Sampling and The model parameters are encoded 1-bit encoding (as in DM), the Transmission and sent to the receiver. bit rate Rb = 2B bps. The receiver utilizes the model With Model Based Coders, It is possible parameters and reconstructs the to reduce the Transmission bit rate Rb to speech using the impulse sequence, less than 2nB bps. generated at the receiver. The methods for the model parameter Human voice is modelled as an output of identification are many and all utilize a system driven with impulses. the inherent correlation in the speech.. The system is characterized by feedback LPC methods are popular and can give and feed-forward parameters. bit rates as low as 2.4 kbps. 72 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
