Communication Theory

Transcription

1 Communication Theory Adnan Aziz Abstract We review the basic elements of communications systems, our goal being to motivate our study of filter implementation in VLSI. Specifically, we review some basic definitions related to signals and systems, and then illustrate the central concepts of modulation and channels, using wireless communications as the motivating example. 1 Signals and Systems Intuition: signals convey information; systems transform signals. Mathematically, a signal is a function mapping a domain into a range (in our context, the domain will be time, either continuous or discrete). A system is a function that takes input signals and returns output signals. As an example, a finite state machine is a system, and its input sequences and output sequences are the signals. For signals that take numerical values, it s often convenient to map the signal into the frequency domain, that is represent it as a sum of sinusoids. This is because for a very important class of systems the Linear Time Invariant systems knowing their behavior on sinusoidal inputs suffices to know there behavior on all inputs. Many signals originating in nature are random, and to understand these, we first review the concept of a random variable. A random variable is a function from a set of random events to the real numbers. For the random variable X, the probability distribution function F X completely spells out its behavior; F X (α) = P r(x α). There is also the related notion of a probability density function (pdf) f X (α), the derivative of F X (α) with respect to α. 1 For any random variable X, one can talk of its mean m X, also denoted by E[X], defined to be + f X(α) dα. 2 That there are RVs for which the integral/sum does converge, but we won t worry about that for now. Analogous to the mean, one can define the variance, which is just the mean of X 2, and the standard deviation σ X given by σx 2 = (E[X 2 ] (E[X] 2 ). 1 We are assuming that the distribution function is continuous and differentiable; this is not the case when for example the underlying set of events is finite. When the random variable takes a discrete set of values, the probability mass function plays the analog of the pdf. 2 For discrete RVs, the integral is replaced by a sum. 1

2 Having reviewed RVs, we can now talk about random signals a random signal is a function whose range is a set of random variables. Suppose S is a continuous-time random signal (i.e., its domain is the set of real numbers); then for any time t, S(t) is a random variable, and we can talk of its mean, variance etc. In the special case that the distribution functions for S(t) are identical for all t, the signal is said to be stationary; in particular this means that its mean at any time is the same. (In practice, all that s needed is for signals to be stationary for long enough.) A stationary signal is ergodic if the average computed over any trace is the same as the mean at any time. Lee and Varaiya have a wonderful text Structure and Interpretation of Signals and Systems, Addison-Wesley, 2003; it s far more accessible than most books which are very math-focused. 2 Modulation We will motivate the need for studying VLSI filtering (and later error correcting codes) by reviewing some of the basic elements of digital wireless communication. Figure 1: A communication system The key idea of digital modulation is to put bits onto a high frequency carrier that the backend sends onto to the antenna; demodulation is the inverse process. Naturally, we can ask Why don t we just send on-off pulses or +A/ A pulses with duration T, like in wired Ethernet? Why don t we send +A, +0.5A, 0.5A, A and convey more information? The choice of modulation scheme influences bit-error-rate (BER), bandwidth, and transceiver complexity/cost. For any modulator, demodulation can be performed in a coherent or incoherent manner coherent means that Rx has exact knowledge of carrier phase reference ( locked in ). We ll assume coherent demodulation. 2.1 BFSK Binary Frequency Shift Keying transmit frequency f 1 for logic 1, f 0 for logic 0. Demodulator correlate with A cos ω 0 t A cos ω 1 t, average over a bit duration. Assume T b 0 cos ω 0 t cos ω 1 t dt = 0, which holds if f 0 and f 1 are multiples of 1/T b. To demodulate, note that for logic 0: s 0 (T b ) = Tb 0 A cos ω 0 t dt = A 2 T b /2 2

3 Figure 2: BFSK modulator Figure 3: BFSK Demodulator and for logic 1: s 1 (T b ) = Tb 0 A cos ω 1 t dt = A 2 T b /2 So we just compare s 0 (T b ) with 0, to determine if a 0 or a 1 was transmitted. 2.2 BPSK Binary Phase Shift Keying change carrier phase. s 0 (t) = A cos ω 0 t s 1 (t) = A cos(ω 0 t + π) = cos ω 0 t Figure 4: BPSK modulator 3

4 Figure 5: BPSK demodulator 2.3 QPSK, OQPSK, MSK 2.4 QPSK Quadrature Phase-Shift Keying Send one of 4 possible symbols: s 00 (t) = A cos ω c t A sin ω c t s 01 (t) = A cos ω c t + A sin ω c t s 10 (t) = A cos ω c t A sin ω c t s 11 (t) = A cos ω c t + A sin ω c t Figure 6: QPSK Demodulator pair of correlators, and pair of coherent reference signals. Figure 7: QPSK demodulator Clearly, different modulation schemes differ in terms of implementation. However, it s not clear how they are different in terms of performance? (Or are they? Perhaps there s a conservation of information law, just like for energy and matter...) 4

5 One consideration: what happens when the channel is not perfect but instead has finite bandwidth (analogous to a trivial RC circuit)? For QPSK, there are large phase changes at the end of each symbol, which means that the signal has high bandwidth. When the channel has finite bandwidth, individual symbols blur and overlap with each other (the technical term is ISI inter-signal interference) OQPSK Offset Quadrature Phase-Shift Keying One way to reduce ISI in QPSK is to avoid simultaneous transitions in waveforms at nodes A and B, thereby reducing signal bandwidth. This is the idea behind OQPSK. Figure 8: OQPSK 2.5 MSK Minimum Shift Keying Another way to avoid large phase changes at the end of each symbol is to adopt a modulation scheme that has continuous phase shift. MSK is such a scheme it can be derived from OQPSK by applying half-sinusoids instead of rectangular pulses to represent the levels that are multiplied by the carriers. Figure 9: MSK 5

6 Now ω 1 is taken to be π/(2t b ); with this it can be shown that S MSL (t) exhibits no abrupt change in phase, i.e., from the frequency domain point of view, MSK signals have a sharper decay in their spectrum, which introduces less distortion when passing through finite bandwidth channels. Question: smoothing seems like a great idea, it saves you signal bandwidth (which is hugely expensive). What s the catch? 3 Channels 3.1 Noisy Channels Channels don t just introduce ISI; they also add noise, which is random. 3 The classical model for channel noise is the AWGN model a noise signal N(t) is added to the signal sent from the transmitter, and n(t) has a Gaussian distribution, and for all t 1 t 2, n(t 1 ) and n(t 2 ) are statistically independent. The addition of noise to the transmitted signal means that there is a finite probability that the demodulator returns the wrong result. Intuitively, how does the probability of error change with the signal amplitude, noise level, transmission rate? It is not very difficult to compute the bit error rates for simpler modulation schemes all that s needed is a little calculus. The errors are often expressed in terms of the error function defined by Q(x) = + x 1/ 2πe x2 dx. For example, for BFSK, P e = Q( (SNR (f N /R b )), where SNR is the signal to noise ratio, f N is the effective noise bandwidth, and R b = 1/T b is the symbol transmission rate. For BPSK, the BER P e = Q( (2 SNR (f N /R b )). (Does this automatically mean BPSK is worse than BFSK?) Let E b and N 0 denote signal-energy-per-bit and noise-power-spectral-density. Note that E b /N 0 = SNR (f N /R b ); hence the bit-error-rate for BFSK and BPSK can be written as Q( (E b /N 0 ) and Q( (2E b /N 0 ). 3.2 Finite Channel Bandwidth One way to counter ISI is to change the pulse shape inside the modulator using a filter (called the transmit filter). Specifically, by sending sinc(t/t b ) instead of a square pulse, there is no ISI. 4 However, the sinc pulse makes it harder to recover timing, and a filter with such sharp transitions is infeasible to build. Consequently, there are other pulses that people have adopted, e.g., the raised-cosine pulse given by p(t) = sinc(t/t b ) cos(παt/t b )/(1 (2αt/T b ) 2 ), where α [0, 1] is the roll-off factor. The raised-cosine does avoid ISI, and is band-limited, but has a higher bandwidth than the ideal (which is 1/T b ). Another pulse that does not completely avoid ISI, but has less bandwidth is the Gaussian pulse given by p(t) = π/αe ( π2 t 2 /α 2). Here α = /B where B is the 3dB bandwidth of the 3 This noise comes from a number of sources, including for example thermal noise in the electronics. 4 The sinc function is sin x/x. 6

7 filter. When applied to MSK, the resulting scheme is GMSK this is for example the scheme use in the DECT (digital enhanced cordless telecommunications) standard. 3.3 Multipath So far, we ve only discussed two nonidealities: AWGN and finite bandwidth. For wireless channels, there are other major problems: path loss, multipath fading, and Doppler shift. Path loss attenuates the transmitted signal, and sets an upper bound on the signal strength the receiver can expect. Multipath attenuates the signal, but also introduces ISI. Doppler shift introduces phase impairment. 3.4 Path loss Theoretical models exists for path loss, based on antenna theory; one example is the Friis equation: P R /P T = G R G T (c/(4π r f)) 2, where G R and G T and the transmit and receive antenna gains, c is the speed of light, r is the distance between transmit and receive antennas, and f the frequency. The Friis equation is for ideal free-space propagation for an LOS path; empirical models show that the loss increases much faster with distance the path loss exponent is between 3.5 and 5 for outdoor cellular, and 2 and 4 for indoor systems. 3.5 Multipath loss Multipath loss is very complex, leading to both amplitude reduction and ISI Amplitude reduction The amplitude reduction introduced by multipath varies with frequency, so one approach to overcoming multipath is to use frequency diversity. Another approach to coping with amplitude reduction is to use spatial diversity multiple antennas separated by multiples of the transmit frequency ISI ISI can be overcome by equalization. Crudely, equalization consists of compensating for the amplitude and phase distortion of the channel with a filter whose transfer function is the inverse of the channel s transfer function. If the channel is changing, the equalizer has to be adaptive. Known training sequences are sent periodically, and the equalizer adapts itself appropriately. The equalizer can be linear (FIR, lattice, etc.) or nonlinear (decision feedback, max likelihood decision feedback, etc.). 3.6 Doppler shift Doppler shifts are a specific example of the general problem of the channel changing very quickly. An engineering solution to this is to use time diversity, i.e., transmit information over a number of differing time slots, where the separation between slots exceeds the coherence time. 7

8 4 OFDM Orthogonal Frequency Division Multiplexing If the time separation between a symbol and its reflections is significantly less than the symbol duration time, then the impact of ISI is less pronounced. Since we cannot change the time separations without moving the antennas around, an obvious solution is to use a large symbol duration time. This however reduces the number of symbols sent per second, i.e., it reduces the data rate. Since the symbol has a longer duration, the noise tends to average out more, so one can send more information per symbol. The obvious way to do this, e.g., PSK using a large number of values from A to +A doesn t work, since for each additional bit the separation between symbols halves, leading to a high BER. One solution is to divide the available channel spectrum into sub-channels, and use filtering to keep them independent. The requirements on the filters is very stringent, and hard to maintain. OFDM uses a different approach: it uses a number of carriers within the spectrum that have spectral overlap; however, the effect of overlap is mitigated by having the carriers be orthogonal. Specifically, if we are transmitting via QPSK on frequency f 0 and also QPSK on f 1, with a symbol duration time T, and the orthogonality requirement T 0 cos 2πf 0 t cos 2πf 1 t dt = 0 is met, then even if the two transmissions overlap in terms of their spectral content, the net effect of one transmission over the other over time T is zero. OFDM is this idea with many (52 (802.11a), 8192 (DVB-T)) frequencies, all of which pair-wise satisfy the orthogonality requirement. (This is easier to achieve than it sounds: use frequencies F 0, f 0 +, f 0 + 2,..., f 0 (n 1), and have = m/t, where m is any integer.) They key advantage of OFDM is that you can overcome the effect of ISI. However, every silver lining has a cloud: It s very hard to ensure the orthogonality requirement: everything from phase noise in the PLLs to the impact of finite-precision arithmetic impacts it The hardware cost is nontrivial e.g., for DVBT implemented directly, need 8192 QPSK transceivers View symbol generation as an inverse DFT operation: can use FFT techniques! 4.1 OFDM Example Given 4 subcarriers, each modulated using BPSK. 8

9 Figure 10: 24 bits of data to be transmitted Figure 11: Carrier 1 waveform over 6 symbol periods 5 Examples of standards 5.1 DECT Example a Example Figure 12: Carrier 2 waveform over 6 symbol periods 9

10 Figure 13: Carrier 3, 4 waveform over 6 symbol periods Figure 14: Concurrent view of all carriers Figure 15: Final aggregate waveform this is what is mixed up and then pushed by the PA onto the antenna 10

11 Figure 16: We are transmitting the sum of several sinusoids. Therefore, can view data to be transmitted as a specification on the spectrum Figure 17: Frequency synthesis naturally performed via IFFT Cell Range m Frequency Range MHz Carrier Spacing MHz peak Channels/carrier 2 x 12 Duplex method TDD using two slots on the same RF carrier Channelization TDMA/FDMA Speech Coding 32 kbps ADPCM Modulation GMSK (BT b = 0.3) Gross data rate Mbps BER < 10 3 Max transmitted power 250 mw Figure 18: Digital Enhanced Cordless Telecommunications 11

12 Data Rate 6, 9, 12, 18, 24, 36, 48, 54 Mbps Modulation BPSK, QPSK, 16-QAM, 64-QAM Coding Rates 1/2, 9/16, 2/3, 3/4 Number of Subcarriers 52 Number of Pilot Tones 4 OFDM Symbol Duration 4 µsec Guard Interval 800 ns Subcarrier Spacing khz Signal Bandwidth MHz Channel Spacing 20 MHz Figure 19: a standard 12