Square Root Raised Cosine Filter

Transcription

1 Wireless Information Transmission System Lab. Square Root Raised Cosine Filter Institute of Communications Engineering National Sun Yat-sen University

2 Introduction We consider the problem of signal design when the channel is band-limited to some specified bandwidth of W Hz. The channel may be modeled as a linear filter having an equivalent low-pass frequency response C( f ) that is zero for f >W. Our purpose is to design a signal pulse g(t) in a linearly modulated signal, represented as v t = I g t nt ( ) ( ) n that efficiently utilizes the total available channel bandwidth W. When the channel is ideal for f W, a signal pulse can be designed that allows us to transmit at symbol rates comparable to or exceeding the channel bandwidth W. When the channel is not ideal, signal transmission at a symbol rate equal to or exceeding W results in inter-symbol interference (ISI) among a number of adjacent symbols. 2 n

3 Characterization of Band-Limited Channels For our purposes, a band-limited channel such as a telephone channel will be characterized as a linear filter having an equivalent low-pass frequency-response characteristic C( f ), and its equivalent low-pass impulse response c(t). Then, if a signal of the form ( ) ( ) j 2 ft c s t = Re v t e π is transmitted over a band-pass telephone channel, the equivalent low-pass received signal is () t = v( τ )( c t τ ) dτ + z() t = v() t c() t z() t r l + where z(t) denotes the additive noise. 3

4 Characterization of Band-Limited Channels Alternatively, the signal term can be represented in the frequency domain as V( f )C( f ), where V( f ) = F[v(t)]. If the channel is band-limited to W Hz, then C( f ) = 0 for f > W. As a consequence, any frequency components in V( f ) above f = W will not be passed by the channel, so we limit the bandwidth of the transmitted signal to W Hz. Within the bandwidth of the channel, we may express C( f ) as jθ f C f = C f e ( ) ( ) ( ) where C( f ) : amplitude response. θ( f ): phase response. The envelope delay characteristic: τ ( f ) = 1 2 dθ π ( f ) df 4

5 Characterization of Band-Limited Channels A channel is said to be nondistorting or ideal if the amplitude response C( f ) is constant for all f W and θ( f ) is a linear function of frequency, i.e., τ( f ) is a constant for all f W. If C( f ) is not constant for all f W, we say that the channel distorts the transmitted signal V( f ) in amplitude. If τ( f ) is not constant for all f W, we say that the channel distorts the signal V( f ) in delay. As a result of the amplitude and delay distortion caused by the nonideal channel frequency-response C( f ), a succession of pulses transmitted through the channel at rates comparable to the bandwidth W are smeared to the point that they are no longer distinguishable as well-defined pulses at the receiving terminal. Instead, they overlap, and thus, we have inter-symbol interference (ISI). 5

6 Characterization of Band-Limited Channels Fig. (a) is a band-limited pulse having zeros periodically spaced in time at ±T, ±2T, etc. If information is conveyed by the pulse amplitude, as in PAM, for example, then one can transmit a sequence of pulses, each of which has a peak at the periodic zeros of the other pulses. 6

7 Characterization of Band-Limited Channels However, transmission of the pulse through a channel modeled as having a linear envelope delay τ( f ) [quadratic phase θ( f )] results in the received pulse shown in Fig. (b), where the zerocrossings that are no longer periodically spaced. 7

8 Characterization of Band-Limited Channels A sequence of successive pulses would no longer be distinguishable. Thus, the channel delay distortion results in ISI. It is possible to compensate for the nonideal frequency-response of the channel by use of a filter or equalizer at the demodulator. Fig. (c) illustrates the output of a linear equalizer that compensates for the linear distortion in the channel. 8

9 Signal Design for Band-Limited Channels The equivalent low-pass transmitted signal for several different types of digital modulation techniques has the common form v () t = I g( t nt ) n= 0 where {I n }: discrete information-bearing sequence of symbols. g(t): a pulse with band-limited frequencyresponse G( f ), i.e., G( f ) = 0 for f > W. This signal is transmitted over a channel having a frequency response C( f ), also limited to f W. The received signal can be represented as () ( ) () rl t = I nh t nt + z t n=0 where h() t = g( τ )( c t τ ) dτ and z(t) is the AWGN. 9 n

10 Suppose that the received signal is passed first through a filter and then sampled at a rate 1/T samples/s, the optimum filter from the point of view of signal detection is one matched to the received pulse. That is, the frequency response of the receiving filter is H*( f ). We denote the output of the receiving filter as where Signal Design for Band-Limited Channels y () t = I x( t nt ) + v() t n= 0 n x(t): the pulse representing the response of the receiving filter to the input pulse h(t). v(t): response of the receiving filter to the noise z(t). 10

11 If y(t) is sampled at times t = kt +τ 0, k = 0, 1,, we have y ( kt τ ) y = I x( kt nt + τ ) + v( kt + ) or, equivalently, Signal Design for Band-Limited Channels + = 0 k n 0 τ 0 n 0 y = I x + v, k = 0,1,... k n k n k n= 0 where τ 0 : transmission delay through the channel. The sample values can be expressed as 1 yk = x 0 Ik + Inx k n + vk, k = 0,1,... x 0 n= 0 n k 11

12 We regard x 0 as an arbitrary scale factor, which we arbitrarily set equal to unity for convenience, then where n= 0 n k I n x k Signal Design for Band-Limited Channels y = I + I x + k k n= 0 n k n k n I k : the desired information symbol at the k-th sampling instant. n: ISI v k : additive Gaussian noise variable at the k-th sampling instant. v k 12

13 Signal Design for Band-Limited Channels The amount of ISI and noise in a digital communication system can be viewed on an oscilloscope. For PAM signals, we can display the received signal y(t) on the vertical input with the horizontal sweep rate set at 1/T. The resulting oscilloscope display is called an eye pattern. Eye patterns for binary and quaternary amplitude-shift keying (or PAM): The effect of ISI is to cause the eye to close. Thereby, reducing the margin for additive noise to cause errors. 13

14 Signal Design for Band-Limited Channels Effect of ISI on eye opening: ISI distorts the position of the zero-crossings and causes a reduction in the eye opening. Thus, it causes the system to be more sensitive to a synchronization error. 14

15 Signal Design for Band-Limited Channels For PSK and QAM, it is customary to display the eye pattern" as a two-dimensional scatter diagram illustrating the sampled values {y k } that represent the decision variables at the sampling instants. Two-dimensional digital eye patterns." 15

16 Design of Band-Limited Signals for No ISI The Nyquist Criterion Assuming that the band-limited channel has ideal frequencyresponse, i.e., C( f ) = 1 for f W, then the pulse x(t) has a spectral characteristic X( f ) = G( f ) 2, where x () t X ( f ) We are interested in determining the spectral properties of the pulse x(t), that results in no inter-symbol interference. Since y = I + I x + v k = W W k n= 0 n k 16 n j 2πft k n the condition for no ISI is 1 0 x( t = kt) xk = 0 0 e df k ( k = ) ( k ) (*)

17 Design of Band-Limited Signals for No ISI The Nyquist Criterion Nyquist pulse-shaping criterion (Nyquist condition for zero ISI) The necessary and sufficient condition for x(t) to satisfy ( ) x nt ( n = ) ( n ) 1 0 = 0 0 is that its Fourier transform X(f ) satisfy m= X ( f + m T ) = T 17

18 Proof: Design of Band-Limited Signals for No ISI The Nyquist Criterion In general, x(t) is the inverse Fourier transform of X( f ). Hence, () t X ( f ) At the sampling instant t = nt, x x = j e 2 πft df ( nt ) X ( f ) = j e 2 πfnt df 18

19 Design of Band-Limited Signals for No ISI The Nyquist Criterion Breaking up the integral into integrals covering the finite range of 1/T, thus, we obtain ( 2m+ 1) 2T j2π fnt x( nt) = X ( f ) e df ( 2m 1) 2T m= 12T j2 π f ' nt = ( ' + ) 12T m= 12T = 12T + m= = 12T 12T ( ) where we define B( f ) as X f m T e df ( ) j2π fnt X f m T e df j2π fnt B f e df B ( f ) = X ( f + m T ) m= ' (1) m f = f ' + T 2m 1 2m+ 1 f :, 2T 2T 1 1 f ':, 2T 2T df = df ' 19

20 Design of Band-Limited Signals for No ISI The Nyquist Criterion Obviously B( f ) is a periodic function with period 1/T, and, therefore, it can be expanded in terms of its Fourier series coefficients {b n } as where j 2 ( ) B f = bne π n= ( ) nft 12T 2π nft bn = T B f e df 12T (2) Comparing (1) and (2), we obtain b n = Tx ( nt ) Recall that the conditions for no ISI are (from ): ( k = ) ( k ) 1 0 x( t = kt) xk =

21 Design of Band-Limited Signals for No ISI The Nyquist Criterion Therefore, the necessary and sufficient condition for y = I + I x + v k k n= 0 n k to be satisfied is that T ( n = 0) bn = 0 ( n 0) which, when substituted into B f j nft = b e 2π n, yields n B ( f ) = T k n ( ) = n k or, equivalently m= X ( f + m T ) = T This concludes the proof of the theorem. 21

22 Design of Band-Limited Signals for No ISI The Nyquist Criterion Suppose that the channel has a bandwidth of W. Then C( f ) 0 for f > W and X( f ) = 0 for f > W. When T < 1/2W (or 1/T > 2W) ( ) ( ) + Since B f = X f + n T consists of nonoverlapping replicas of X( f ), separated by 1/T, there is no choice for X( f ) to ensure B( f ) T in this case and there is no way that we can design a system with no ISI. 22

23 Design of Band-Limited Signals for No ISI The Nyquist Criterion When T = 1/2W, or 1/T = 2W (the Nyquist rate), the replications of X( f ), separated by 1/T, are shown below: In this case, there exists only one X( f ) that results in B( f ) = T, namely, T ( f < W) X ( f ) = 0 ( otherwise) which corresponds to the pulse sin ( πtt) πt xt () = sinc πtt T 23

24 Design of Band-Limited Signals for No ISI The Nyquist Criterion The smallest value of T for which transmission with zero ISI is possible is T = 1/2W, and for this value, x(t) has to be a sinc function. The difficulty with this choice of x(t) is that it is noncausal and nonrealizable. A second difficulty with this pulse shape is that its rate of convergence to zero is slow. The tails of x(t) decay as 1/t; consequently, a small mistiming error in sampling the output of the matched filter at the demodulator results in an infinite series of ISI components. Such a series is not absolutely summable because of the 1/t rate of decay of the pulse, and, hence, the sum of the resulting ISI does not converge. 24

25 Design of Band-Limited Signals for No ISI The Nyquist Criterion When T > 1/2W, B( f ) consists of overlapping replications of X(f) separated by 1/T: In this case, there exist numerous choices for X( f ) such that X( f ) such that B( f ) T. 25

26 Design of Band-Limited Signals for No ISI The Nyquist Criterion A particular pulse spectrum, for the T > 1/2W case, that has desirable spectral properties and has been widely used in practice is the raised cosine spectrum. Raised cosine spectrum: 1 β T 0 f 2T T πt 1 β 1-β 1+ β Xrc ( f ) = 1+ cos f f 2 β 2T 2T 2T 1+ β 0 f > 2T β: roll-off factor. (0 β 1) 26

27 Design of Band-Limited Signals for No ISI The Nyquist Criterion The bandwidth occupied by the signal beyond the Nyquist frequency 1/2T is called the excess bandwidth and is usually expressed as a percentage of the Nyquist frequency. β= 1/2 => excess bandwidth = 50 %. β= 1 => excess bandwidth = 100%. The pulse x(t), having the raised cosine spectrum, is x () t = = sin πt sin ( πt T ) cos( πβt T ) 1 ( πt T ) 2 4β t cos 1 4β x(t) is normalized so that x(0) = 1. c T ( πβt T ) 2 2 t 2 T 2 T 2 27

28 Design of Band-Limited Signals for No ISI The Nyquist Criterion Pulses having a raised cosine spectrum: Forβ= 0, the pulse reduces to x(t) = sinc(πt/t), and the symbol rate 1/T = 2W. When β= 1, the symbol rate is 1/T = W. 28

29 Design of Band-Limited Signals for No ISI The Nyquist Criterion In general, the tails of x(t) decay as 1/t 3 for β> 0. Consequently, a mistiming error in sampling leads to a series of ISI components that converges to a finite value. Because of the smooth characteristics of the raised cosine spectrum, it is possible to design practical filters for the transmitter and the receiver that approximate the overall desired frequency response. In the special case where the channel is ideal, i.e., C( f ) = 1, f W, we have X f G f G f ( ) ( ) ( ) rc = where G T ( f ) and G R ( f ) are the frequency responses of the two filters. T R 29

30 Design of Band-Limited Signals for No ISI The Nyquist Criterion If the receiver filter is matched of the transmitter filter, we have X rc ( f )= G T ( f ) G R ( f ) = G T ( f ) 2. Ideally, G ( ) T 2 ft0 ( f ) X ( f ) e j π = rc and G R ( f ) = G T f, where t 0 is some nominal delay that is required to ensure physical realizability of the filter. Thus, the overall raised cosine spectral characteristic is split evenly between the transmitting filter and the receiving filter. An additional delay is necessary to ensure the physical realization of the receiving filter. 30

31 Wireless Information Transmission System Lab. Implementation of Square Root Raise Cosine Filter Institute of Communications Engineering National Sun Yat-sen University

32 Simplified System Architecture Transmitter Receiver SRRC-TX DAC LPF DAC SRRC-RX 32

33 Design of SRRC Operation of the SRRC-TX module includes: Over-sampling Over-sampling rate is a system design issue (performance vs. cost). For a over-sampling rate = 4, we pad three zeros between each sample. Finite-impulse-response (FIR) filter The FIR filter acts as a low pass filter. The FIR filter is a square root raised cosine filter with rolloff factor β=

34 Detail of FIR Filter Module Input h(0) h(1) h(2) h(n/2) 0 34

35 Raised Cosine Filter Raised cosine spectrum: 1 β T 0 f 2T T πt 1 β 1-β 1+ β Xrc ( f ) = 1+ cos f f 2 β 2T 2T 2T 1+ β 0 f > 2T The pulse x(t), having the raised cosine spectrum, is ( ) ( ) 35 ( ) sin πtt cos πβtt cos πβtt x() t = = sin c( π t T) πt T 1 4β t T 1 4β t T

36 Square Root Raised Cosine Filter The cosine roll-off transfer function can be achieved by using identical square root raised cosine filter X at the rc ( f ) transmitter and receiver. The pulse SRRC(t), having the square root raised cosine spectrum, is t t t sin π ( 1 β) + 4β cos π ( 1+ β) TC TC TC SRRC () t = 2 t t π 1 4β T C T C where TC is the inverse of chip rate ( µ s) and β =

37 Square Root Raised Cosine Filter Because SRRC(t) is non-causal, it must be truncated, and pulse shaping filter are typically implemented for ±6T C about the t = 0 point for each symbol. For an over-sampling rate of 4, n is equal to