CT-516 Advanced Digital Communications

CT-516 Advanced Digital Communications Yash Vasavada Winter 2017 DA-IICT Lecture 17 Channel Coding and Power/Bandwidth Tradeoff 20 th April 2017

Power and Bandwidth Tradeoff (for achieving a particular data rate) 1 Transmitter designer s objective: use as little power as possible to deliver the target data rate In general, communication system design is constrained by the power. Reduction in the power requirement is always of system benefit and of economic value Maximum power that can be delivered is limited by the amplifier used. More expensive the amplifier, the greater the max power but also greater (often prohibitively) the cost There is a way to use less power to deliver the target data rate: increase the bandwidth 2

Power and Bandwidth Tradeoff (for achieving a particular data rate) 2 System designer s objective: use as little bandwidth as possible to deliver the target data rate In general, communication system design is constrained by the bandwidth. Reduction in the bandwidth requirement is always of system benefit and of economic value Available bandwidth is limited by the regulatory agency. High bandwidth implies greater cost of purchase (like buying a bigger real estate) There is a way to use less bandwidth to deliver the target data rate: increase the power Brings us back to Square One! 3

Three Critical System Parameters Following are three critical parameters of a digital communication system 1. Data Rate: Need is to maximize the data rate that can be delivered over the DIGICOM system 2. Power 3. Bandwidth Achieved data rate increases with power and it increases with bandwidth. In reality, both these resources are limited Data rate achieved is a function of power and bandwidth. We look at this relationship next 4

Data Rate, Power and Bandwidth Considerations in a DigiComm System Transmitted power is Attained bandwidth is determined by the design of modem determined by the HPA used Allotted bandwidth is determined by the spectrum granted by the regulatory bodies (e.g., FCC, ITU) Information Bits from the User Decoded Information Bits to the User Channel Encoder Channel Encoder Modulator Demodulator Freq. Upconverter Band Pass Filter (BPF) Reference Oscillator Band Pass Filter (BPF) High Powered Amplifier (HPA) Low Noise Amplifier (LNA) Diplexer Transmit and Received Signals Data rate is as required by the user (e.g., a video download/upload will have a much greater data rate requirement compared to transfer of emails) Tuning Control Control Subsystem Antenna Control 5

Amplitude, Power, and Energy Energy E in Joules is power P Watts integrated over time T For the following waveform, energy per symbol: E S A 2 T S = P T S = P/R S Here, T S is the symbol duration in seconds and R S is the symbol rate in symbols/sec Amplitude A Binary Symbol Stream with Rectangular Pulse Shape A A Time Symbol Duration T S 6

Amplitude, Power and Energy A rough analogy and a conceptual (mathematically inexact) visualization of the dual relationship between the power and energy in time-frequency domains Power, Watts Time Domain Energy, Joules Frequency Domain P E Area: P T S = P/R S equals the energy E over time period T S Area: E R S = E/T S equals the power P over bandwidth R S Time Frequency Span T S = 1/R S Span R S = 1/T S Thus, two ways of measuring the quality or fidelity of the received signal compared to the noise Ratio of power levels of signal and noise (level ratio in time domain, area ratio in frequency domain) Ratio of energy levels of signal and noise (level ratio in frequency domain, area ratio in time domain) 7

Example: White Noise The noise affecting the satellite communication link is often modeled as spectrally white Analogous to the spectrum of the white light, the noise is typically observed to have all the frequency components that are statistically equally powered (i.e., power spectral density or PSD equals a constant N 0 ) Power Spectral Density N 0 Power Spectral Density of White Noise Ideal (Brickwall) Filter with Bandwidth B Filter Bandwidth B Frequency At the receiver, the received signal plus the noise is filtered by a filter with bandwidth B. The noise power at the filter output is, therefore, P N = N 0 B 8

Symbol Rate and Bandwidth Square wave of width T seconds has a spectrum that spans over 1 T Time Domain Frequency Domain Communication systems rarely use square wave. A common practice is Root Raised Cosine (RRC) pulse shape 9

Symbol Rate and Bandwidth Raised Cosine Filters with Different Roll-off Factors α Time Domain Frequency Domain If α = 0, it is seen that a symbol rate of T S translates to a bandwidth of B = 1 T S = R S In general, the rule of thumb, for arbitrary α is as follows: B = R S (1 + α) 10

Symbol Rate and Information (Bit) Rate How is the symbol rate R s related to the bit rate in a digital communication system? To answer this question, we need to consider the block diagram of the transmitter A Simplified Diagram of the Transmitter of a Digital Communication System Information Bit Stream Bit Rate: R b bits/sec Channel Encoder Code Rate: r Encoded Bit Stream Modulator (Bit to Symbol Mapper) Symbol Stream Symbol Rate: R S symbols/sec 11

Simplified Versus Actual Transmitter Simplified: Information Bit Stream Bit Rate: R b bits/sec Channel Encoder Code Rate: r Encoded Bit Stream Modulator (Bit to Symbol Mapper) Symbol Stream Symbol Rate: R S symbols/sec Detailed (for DVB-S2 system): 12

Symbol Rate and Bit Rate Channel Encoder: introduces redundancy in the information bit stream to better protect against the noise encountered in the transmission channel. Every block of, say, N information bits is transformed to an encoded block of N r r, 0 < r 1, is the coding rate. N encoded bits, where Protection against the noise is achieved at the cost of a precious measure. Incoming rate of R b is (artificially) inflated to R b /r bits/sec Modulator: an M-ary modulator takes a group of k = log 2 M symbols and maps each possible group to a unique symbol Every block of k encoded bits at the output of the channel encoder is transformed to a modulated symbol Therefore, the symbol rate becomes R s = R b r k 13

Symbol Rate and Information (Bit) Rate Now, we can answer the question: How is the symbol rate R s related to the bit rate in a digital communication system? Symbolic Example Unit Information Bit Rate R b 100 Mbps Code Rate r 2/5 none Encoded Bit Rate R b /r 250 Mbps Modulation Type k = log 2 M 5 none Symbol Rate R S = 50 Msps Symbol Rate R S = R b r log 2 M or Bit Rate R b = R S r log 2 M = R b r k B 1+α r log 2 M 14

Symbol Rate and Information (Bit) Rate Consider the formulation of bit rate R b = R S r log 2 M = What happens if: 1. Code rate r is reduced by a factor of x? Required power reduces (we will see this shortly) B 1+α r log 2 M However, bandwidth B has to be increased by x to maintain the same code rate 2. Modulation size M is increased by a factor of x? Required bandwidth B to deliver the data rate R b reduces by x However (as we will see shortly), the required power increases This is the power-bandwidth tradeoff mentioned earlier In (1) above, power is reduced but we ended up consuming more bandwidth In (2) above, the bandwidth is reduced, but we ended up consuming more power 15

M and k for Different Modulation Schemes Bits to Symbol Mapping is performed by the modulator Modulator converts the real-valued bit stream to (typically) a complex-valued symbol stream Bits are binary (take only two values); symbols are M-ary (take any one of, typically, M = 2 k values) A serial bit stream is converted to a parallel bit stream where each of the parallel branches comprises of a block of k sequential bits. Each such block is transformed to one of M symbols QPSK: (M, k = 4,2) 8-PSK: (M, k = 8,3) 16-APSK: (M, k = 16,4) 16-QAM: (M, k = 16,4) 64-QAM: (M, k = 64,6) 16

Channel Encoding: Why Is It Needed? Channel encoding scheme makes the transmitter send more bits than needed However, extra bits provided parity checks and they help the receiver recover from the errors in the received message An extremely simple example of channel encoding: can undo one bit of error every three bits Rate r = 1/3 Incoming Bit Encoded Bits; Rate r = 1/3 1 1 1 1 Received Bits 1 1 1 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 Decoded Bit 0 0 0 0 0 1 17

SNR and Bit Error Rate Effect of error control coding is to reduce the required E b /N 0 to achieve the same bit error rate This directly allows a power saving However, this power saving is achieved at the cost of an increased bandwidth consumption as noted earlier Potential performance of LDPC R = ½ BPSK/QPSK Coding Gain Uncoded Performance https://commons.wikimedia.org/w/index.php?curid=21966 75 18

SNR and Bit Error Rate: Recent Advancement in Channel Encoding/Decoding Observation 1: As M, required E S /N 0 and hence, required power As r, required power Observation 2: AWGN channel turns into a nearly on-off channel dictated by Es/No threshold Error rate falls off a cliff above the threshold; near zero errors above the threshold, near certainty of the error below the threshold No need (ideally) to specify a packet error requirement; in reality a performance floor not seen here is likely to surface; performance cannot be improved beyond some very low packet error rate Assumes a long length packet (64800 bits) [2]; shorter length packets may incur about 1 db penalty as seen in comparison with the previous and the next slides 19

SNR and Bit Error Rate: Recent Advancement in Channel Encoding/Decoding Conventional channel coded utilized either block codes such as Reed-Solomon or BCH codes, or Convolutional codes, or a concatenation of the two. In 1990 s, two new codes, LDPC and Turbo codes, improved upon the coding technology. Specifically, as the packet length of LDPC and Turbo codes increases 1. The frame error rate (FER) of the encoded packet reduces This is opposite of the behavior of the conventional codes, whose FER typically degrades with increasing packet size However, since the packet length N has to be made large for LDPC or Turbo codes, each instance of packet error carries an increased consequence 2. The FER curves start exhibiting a cliff This is in contrast to the waterfall-shaped FER performance curves for the conventional code The LDPC/Turbo FER cliff can be made more vertical by increasing the decoder complexity (# of iterations) o No such complexity-performance trade typically exists for the conventional codes Thus, with LDPC/Turbo codes, for large N values, only the cliff-point Es/No (or threshold) needs to be budgeted for. The channel becomes nearly error-free above the threshold Es/No o o This is unlike the convolutional code for which Es/No has to be tied to a specific FER number A side consequence: if Es/No drops below LDPC/Turbo threshold, the performance may severely degrades 20

Fundamental (Theoretical) Limits System designer s objective: achieve the highest data rate possible System constraints: limited available power and limited available bandwidth Maximum possible data rate R b that can be achieved given the power and bandwidth constraint is bounded by the following Information Theory result (originally developed by Claude Shannon in 1948) that sets the channel capacity C Assumes that the only impairment introduced by the transmission channel is the Additive White Gaussian-distributed Noise. Such a channel is accordingly called the AWGN channel, and the following formula is known as the capacity of the AWGN channel R b C = B log 2 1 + P S P N bits per second 21

Spectral Efficiency Spectral efficiency is a key metric that denotes whether the available bandwidth is utilized efficiently for data transport Defined as the ratio of bit rate to bandwidth: R b B C B = log 2 1 + P S P N in bps/hz Higher value many bits are getting transported for each Hertz of bandwidth Possible when a lot of power is available Low value only a few bits are transported for each Hz of bandwidth Possible when a lot of bandwidth is available Can alternatively be also defined as A function of E S /N 0 : R b B C B = log 2 1 + E S N 0 in bps/hz A function of E b /N 0 : R b C = log B B 2 1 + C E b in bps/hz B N 0 Pulse-shaping factor α taken to be zero in the above Refer to appendix for a detailed conversion from Es/No to Eb/No 22

Shannon Capacity of AWGN Channel: Spectral Efficiency Spectral efficiency in bps/hz as a function of E b /N 0 : C/B in bps/hz R b B C B = log 2 1 + C B E b N 0 in bps/hz Spectral efficiency becomes zero when E b /N 0 is -1.6 db Effect of noise can be completely removed by channel encoder provided E b N 0 > 1.6 db and a lot of bandwidth is available Shannon capacity applies when the transmitted symbols are not constrained to lie on a pre-defined constellation Modulation constrained capacity is smaller than the Shannon capacity and is also shown here E b /N 0 in db 16-PSK 64-QAM 16-APSK 8-PSK QPSK BPSK 23

Shannon Capacity of AWGN Channel: Spectral Efficiency Spectral efficiency in bps/hz as a function of E S /N 0 : C/B in bps/hz R b B C B = log 2 1 + E S N 0 in bps/hz Since E s = E b C N 0 N 0 B, this result is obtained by scaling the X axis of the previous chart by the values along the Y axis 16-APSK 16-PSK 8-PSK QPSK BPSK E S /N 0 in db 24

Shannon Capacity of AWGN Channel: Spectral Efficiency Attainable performance with practical channel coding technology and QPSK modulation As a function of E b /N 0 As a function of E S /N 0 25

Spectral Efficiencies and Required Es/No for Different Modulation and Coding Schemes DVB-S2 Modulation and Coding Bits/Hz Code Rate Bits/Symbol Code Rate Modulation Es/No 0.490 0.25 2 0.250 QPSK -2.35 0.656 0.33 2 0.333 QPSK -1.24 0.789 0.39 2 0.400 QPSK -0.30 0.988 0.49 2 0.500 QPSK 1.00 1.188 0.59 2 0.600 QPSK 2.23 1.322 0.66 2 0.667 QPSK 3.10 1.487 0.74 2 0.750 QPSK 4.03 1.587 0.79 2 0.800 QPSK 4.68 1.654 0.83 2 0.833 QPSK 5.18 1.766 0.88 2 0.889 QPSK 6.20 1.788 0.89 2 0.900 QPSK 6.42 1.980 0.66 3 0.667 8PSK 6.62 2.281 0.76 3 0.750 8PSK 7.91 2.478 0.83 3 0.833 8PSK 9.35 2.646 0.88 3 0.889 8PSK 10.69 2.679 0.89 3 0.900 8PSK 10.98 3.165 0.79 4 0.800 16APSK 11.03 3.300 0.83 4 0.833 16APSK 11.61 3.523 0.88 4 0.889 16APSK 12.89 3.567 0.71 5 0.900 16APSK 13.13 3.951 0.79 5 0.800 32APSK 13.64 4.119 0.82 5 0.833 32APSK 14.28 4.397 0.88 5 0.889 32APSK 15.69 4.453 0.89 5 0.900 32APSK 16.05 26

DVB-S2 versus DVB-S2X 27

DVB-S2 versus DVB-S2X 28

Shannon Capacity of AWGN Channel: Spectral Efficiency In an alternate depiction of Power and Bandwidth trade-off, we consider here a scenario with a constant bit rate of C = 10 bps Code rate r is linearly increased from 0 to 1 Bandwidth B reduction is inversely proportional to this linear increase Noise spectral density N 0 is set to 1/6 W/Hz The attached plot shows the required power and the bandwidth as a function of bps/hz 29

Shannon Capacity of AWGN Channel: Constant SNR Operation Channel Capacity in bits per second If SNR can be held constant, the channel capacity C grows linearly with the bandwidth B = β: For example, if P S P N = γ = 1, C = β log 2 1 + γ = β log 2 1 + 1 = β; i.e., k = 1 Note: since P N = N 0 β, a constant P S P N = γ with increasing β is possible in AWGN channel only if the signal power P S is increased in the same proportion as β. This is often infeasible Normalized channel capacity in bits per second per Hertz An important efficiency measure is how well the allotted bandwidth is being utilized to deliver the data rate. This is called the spectral efficiency and it is given as the ratio of the data rate to the bandwidth and has a unit of bps/hz This plot shows that the spectral efficiency reduces if SNR becomes small 30

Power and Bandwidth Tradeoff: Constant Power Operation Let us assume a power constrained operation Transmit power P is a constant Channel capacity is given as a function of B as the following nonlinear function In this plot, this function is plotted for k = P/N 0 values of 6 db, 0 db, +6 db, +12 db 31

Power and Bandwidth Tradeoff: Constant Bandwidth Operation Let us assume a bandwidth constrained operation: bandwidth B is a constant Channel capacity is given as a function of P as the following nonlinear function In this plot, this function is plotted for k = B values of 1, 2, 4, 8 and 16 Hz with N 0 = 1 W/Hz 32

Power and Bandwidth Tradeoff: Constant Data Rate Operation Given the bandwidth B = β Hz, the following defines the power P(β) as function of β for a given channel capacity C This function is plotted as a function of β for N 0 = 1 Watts/Hz, and for several values of C 33

Appendix 34

Signal to Noise Ratio and Bit Error Rate Performance of communication system is determined by the ratio of the (carrier or) signal power P S = C to the noise power P N = C at the receiver. This is called SNR or C/N. SNR = C N = P S P N There are many different, equivalent, ways of determining how strong the signal is compared to the noise 1. Power ratio (level difference in time domain): SNR 2. Energy ratio (level difference in frequency domain): E S /N 0 and E b /N 0 3. Mixed: C/N 0 How are these related? 35

Signal to Noise Ratio and Bit Error Rate How are the different ways of specifying the strength of the signal relative to the strength of the noise related to each other? 1. As we have seen earlier, signal power P S = E S R S, where E S is the signal energy in dbw/hz 2. As we have noted earlier, the noise power P N = N 0 B, where N 0 is the spectral density of the noise in dbw/hz 3. We have seen that B = 1 + α R S. Also, we have shown that R b = r log 2 k R S. 4. Similarly, it can be shown that E b = E S / r log 2 k 5. Thus, E S R S = E b R b Using the above relationships, we can convert one metric of SNR to another 36

Signal to Noise Ratio and Bit Error Rate Therefore, SNR = C N = P S P N = E SR S N 0 B = 1 B C N 0 = E S 1 + α N 0 = E b R b 1 + α N 0 R S = 1 1 + α r log 2 k E b N 0 Common Measures of Signal Quality and a Table of Conversion Among Them y = c x c SNR E S /N 0 C/N 0 E b /N 0 SNR c = 1 c = 1 + α c = B c = 1 + α r log 2 k x E S /N 0 c = 1 + α 1 c = 1 c = R S c = r log 2 k = R b R S C/N 0 c = B 1 c = R S 1 c = 1 c = R S 1 E b /N 0 c = 1 r log 2 k 1 + α c = 1 r log 2 k = R S R b c = B r log 2 k c = 1 37