Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 1-1 st Semester, 008 Chpater 8 Digital Transmission through Bandlimited AWGN Channels Text. [1] J. G. Proakis and M. Salehi, Communication Systems Engineering, /e. Prentice Hall, 00. 8.1 Digital Transmission through Bandlimited Channels 8. Power Spectral Density of the Baseband Signal 8.3 Signal Design for Bandlimited Channels 8.4 Probability of Error in Detection of Digital PAM 8.5 Digitally Modulated Signals with Memory (partly skipped) 8.6 System Design in the Presence of Channel Distortion (skipped) 8.7 Multicarrier Modulation and OFDM (briefly covered) Digital communication over a channel is modeled as a linear filter with a bandwidth limitation. Bandlimited channels most frequently are encountered in telephone channels, microwave LOS radio channels, satellite channels, and underwater acoustic channels.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - - 1 st Semester, 008 The transmitted signals must be designed to satisfy the bandwidth constraint imposed by the channel, that is, the transmitted signals must be shaped to restrict their bandwidth to that available on the channel. 8.1 Digital Transmission through Bandlimited Channels A bandlimited channel is characterized as a linear filter with impulse response ct () and frequency response C( f ) which is given by jπ f t C( f) c( t) e dt =. (8.1.1) If the channel is a baseband channel that is bandlimited to B c Hz, then C( f ) = 0 for f > Bc. Any frequency components with frequency higher than B c Hz will not be passed by the channel which is bandlimited to W = B Hz as shown in Figure 8.1. c
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 3-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 4-1 st Semester, 008 Let W denote the bandwidth limitation of the signal and the channel. Suppose that the input to a bandlimited channel is a signal gt () t. Then, the output of the channel corresponding to the input g () t is given by ht () = c( τ ) g ( t τ) dτ T = ct () g () t (8.1.) T or, in the frequency domain, we have H ( f) = C( f) G ( f) (8.1.3) T where GT ( f ) is the spectrum (Fourier transform) of the signal gt () t and H ( f ) is the spectrum of ht ( ). T Assume that the signal at the input to the demodulator (that is, at the output of the channel) is corrupted by AWGN which is given by ht ( ) + nt ( ) where nt ( ) is the AWGN. In the presence of AWGN, a demodulator having a filter which is matched to the signal ht ( ) maximizes
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 5-1 st Semester, 008 the SNR at its output. Let the received signal ht ( ) + nt ( ) is passed through the matched filter of which frequency response is given by G ( f) H ( f) e R jπ ft0 = (8.1.4) where t 0 is time delay at which the filter output is sampled. The signal component at the output of the matched filter at the sampling instant t = t0 is given by ys ( t0) = H( f) df = ε h (8.1.5) which is the energy in the channel output ht ( ). The noise component at the output of the matched filter has zero mean and a power-spectral density given by N Sn( f) H( f) 0 =. (8.1.6)
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 6-1 st Semester, 008 Hence, the noise power at the output of the matched filter has a variance σ ( ) n = S n f df N 0 = H ( f) df N0ε = h (8.1.7) The SNR at the output of the matched filter is given by S ε h = N N 0 0ε h ε h = (8.1.8) N 0 which is the same as SNR at the output of the matched filter in Chapter 7 except that the received signal nergy ε h has replaced the transmitted signal energy ε S. Note that the filter impulse response is matched to the signal component ht ( ) in the received signal instead of the transmitted signal.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 7-1 st Semester, 008 Note that to implement the matched filter at the receiver, ht ( ) (or, equivalently, the channel impulse response ct ( ) ) must be known to the receiver. Ex. 8.1.1 The signal g () t is given by T 1 π T gt () t = 1 cos t + T, 0 t T, which is shown in Figure 8.(b). gt 8.(a). () t is transmitted through a baseband channel with frequency-response characteristic as shown in Figure
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 8-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 9-1 st Semester, 008 N 0 The channel output is corrupted by AWGN with power-spectral density. Determine the matched filter to the received signal and the output SNR. Solution The spectrum of the signal is given by T sinπ ft GT ( f) = e π ft (1 f T ) T = sincπ ft e (1 f T ) jπ ft jπ ft of which square, ( ) GT f, is shown in Figure 8.(c). Hence, H ( f) = C( f) G ( f) T GT ( f), f W, = 0, otherwise.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 10-1 st Semester, 008 Then, the signal component at the output of the filter matched to H ( f ) is given by ε h = H ( f) df = W G T ( f ) df W 1 W (sin π ft) = ( π ) f (1 f T ) W df T WT sin πα = dα ( π). WT α (1 α ) The variance of the noise component is given by σ N N0ε = h. W 0 n = T( ) W G f df Hence, the output SNR is given by S = N ε h N 0 0.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 11-1 st Semester, 008 In this example, only a part of the transmitted signal energy is received, since the signal at the input to the channel is not bandlimited. The amount of signal energy at the output of the matched filter depends on the channel bandwidth W when the signal pulse duration is fixed (see Problem 8.1). The maximum value of ε h is obtained by letting W, that is, max ε h = lim ( ) W W = W GT f df GT ( f) df T g 0 T = () t dt. Note that the performance of the system is determined by ε h, the energy in the received signal ht ( ). To maximize the received SNR, we have to make sure that the power-spectral density of the transmitted signal matches the frequency band of the channel.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 1-1 st Semester, 008 8.1.1 Digital PAM Transmission through Bandlimited Baseband Channels Consider the baseband PAM communication system shown by the block diagram in Figure 8.3. The system consists of a transmitting filter having an impulse response g () t, the linear filter channel with AWGN, a receiving filter with impulse response g () t, a sampler that periodically samples the output of receiving filter, and a symbol detector. R T The sampler needs a timing signal whch is extracted from the received signal (see Section 7.8) to serve as a clock to specify the appropriate time instants to sample the output of the receiving filter.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 13-1 st Semester, 008 The input binary data sequence of bit rate R b from the source is subdivided into k -bit symbols and each k -ary symbol a n is mapped into a corresponding amplitude level which modulates the output of the transmitting filter. The baseband signal at the output of the transmitting filter (or at the input to the channel) is given by vt () = ag n T( t nt) (8.1.9) n=
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 14-1 st Semester, 008 where T k = is the symbol interval. R b Note that the symbol rate is given by 1 Rb T =. k The received signal at the demodulator (or the channel output) is given by rt () = aht n ( nt) + nt () (8.1.10) n= where ht ( ) is the impulse response of the cascade of the transmitting filter and the channel, that is, ht () = ct () g () t, T ct () is the impulse response of the channel, and nt () is an AWGN. The received signal is passed through a linear receiving filter with impulse response g () t and frequency response GR( f ). R
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 15-1 st Semester, 008 If gr() t is matched to ht ( ), then its output SNR becomes a maximum at the proper sampling instant. The output of the receiving filter is given by yt () = axt n ( nt) + vt () (8.1.11) n= where x() t = h() t g () t = g () t c() t g () t and R T R vt () = nt () g () t is the additive noise at the output of the receiving filter. R To recover the information symbols { a n }, the output of the receiving filter is sample periodically with the interval of T seconds. The output of the sampler is given by ymt ( ) = axmt n ( nt) + vmt ( ) (8.1.1) n= or, equivalently,
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 16-1 st Semester, 008 y = a x + v m n m n m n= = x a + a x + v (8.1.13) 0 m n m n m n= n m where x = xmt ( ) and m vm = v( mt) for m =,, 1, 0,1,,. The first term, x0a m, on the right-hand side (RHS) of (8.1.13) is the desired symbol a m scaled by the gain parameter x 0. When the receiving filter is matched to the received signal ht ( ), the scale factor is given by 0 = () x h t dt = = H ( f) df W G ( f T ) C ( f ) df W = ε h (8.1.14)
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 17-1 st Semester, 008 as shown in (8.1.4) and (8.1.5). The second term on the RHS of (8.1.13) represents the intersymbol interference (ISI) which is the effect of the other symbols to the desired symbols at the sampling instant t = mt. In general, ISI degrades the performance of the digital communication system. The third term, v m, on the RHS of (8.1.13) is the additive noise and is a zero-mean Gaussian random variable with variance N0ε h σ v = as given by (8.1.7). By appropriate design of the transmitting and receiving filters, it is possible to satisfy the condition x n = 0 for all n 0, so that the ISI term vanishes. In this case, the only term which causes errors in the received digital sequence is the additive noise.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 18-1 st Semester, 008 8.1. Digital Transmission through Bandlimited Bandpass Channels The development given in Section 8.1.1 for baseband PAM is easily extended to carrier modulation via PAM, and QAM and PSK. In a carrier-amplitude modulated signal (or bandpass PAM or ASK), the baseband PAM given by vt ( ) in (8.1.9) modulates the carrier, so that the transmitted signal is given by ut () = vt ()cos π ft, (8.1.15) c which implies that the baseband signal vt ( ) is shifted in frequency by f c. A QAM signal is a bandpass signal. A rectangular QAM signal may be viewed as two amplitude-modulated carrier signals in phase quadrature. That is, the QAM signal is given by ut () = v()cos t π ft+ v()sin t π ft (8.1.16) c c s c
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 19-1 st Semester, 008 where vc() t = ancgt( t nt), n= vs() t = an sgt( t nt) (8.1.17) n= and { a nc} and { a ns} are the two sequences of amplitudes carried on the two quadrature carriers. An equivalent complex-valued baseband signal of the QAM signal is given by vt () = v() t jv() t c s = ( a ja ) g ( t nt) n= nc ns T = ag n T( t nt) (8.1.18) n= where an = anc jans and the sequence { a } is a complex-valued sequence representing the signal points in the QAM signal constellation. n
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 0-1 st Semester, 008 From (8.1.16) and (8.1.18), the corresponding bandpass QAM signal is given by ut =. (8.1.19) j ft c () Re vte () π Similarly to (8.1.19), an equivalent complex-valued baseband (or PSK) signal is given by vt () = ag n T( t nt) (8.1.0) n= and the sequence { a n } takes the value from the set of possible (phase) values m jπ M { e, m= 0,1,, M 1}. All three carrier-modulated signals for PAM, QAM, and PSK can be represented as in (8.1.19) and (8.1.0), where the only difference is in the values of the transmitted sequence { a n }. The signal vt ( ) given by (8.1.19) or (8.1.0) is called the equivalent lowpass signal. In the case of QAM and PSK, the equivalent lowpass signal vt ( ) is a complex-valued baseband signal because the information-bearing sequence { a n } is complex-valued.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 1-1 st Semester, 008 In the case of PAM, the equivalent lowpass signal vt ( ) is a real-valued baseband signal. Suppose that the channel has the impulse response of the equivalent lowpass channel ct ( ). Then, when transmitted through the bandpass channel, the received bandpass signal is given by wt = (8.1.1) j fc t () Re rte () π where rt ( ) is the equivalent lowpass (baseband) signal, which is given by rt () = aht n ( nt) + nt () (8.1.) n= where ht ( ) is the impulse response of the cascade of the transmitting filter and the channel, that is, ht () = ct () g () t, and T nt () is the additive Gaussian noise expressed as an equivalent lowpass (baseband) noise. The received bandpass signal can be converted to a baseband signal by multiplying wt ( ) with the
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - - 1 st Semester, 008 quadrature carrier signals cos π fct and sin π fct and eliminating the double frequency terms by passing the two quadrature components through two separate lowpass filters, as shown in Figure 8.4.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 3-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 4-1 st Semester, 008 Assume each of the lowpass filters in Figure 8.4 has an impulse response gr() t. Then, we can represent the two quadrature components at the outputs of the two lowpass filters as an equivalent complex-valued signal yt () = axt n ( nt) + vt () (8.1.3) n= which is identical to (8.1.11) for the real baseband signal. Hence, the signal design problem for bandpass signals is basically the same as that for baseband signals. 8. Power Spectral Density of Digitally Modulated Signals
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 5-1 st Semester, 008 8..1 Power Spectral Density of the Baseband Signal The equivalent baseband transmitted signal for a digital PAM, PSK, or QAM signal is represented in the general form as vt () = ag n T( t nt) (8..1) n= where { a n } is the sequence of values selected from either a PAM, QAM, or PSK signal constellation corresponding to the information symbols from the source, and gt () t is the impulse response of the transmitting filter. Since the information sequence { a n } is random, vt ( ) is a sample function of a random process Vt. ( ) The mean function of the random variable (or the baseband transmitted signal) vt ( ) is given by E[ Vt ( )] = Ea [ ] g( t nt) n= n T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 6-1 st Semester, 008 = ma gt( t nt) (8..) n= where m a is the mean of the random variable a n. Note that, since although m a is a constant, the term gt ( t nt) in (8..) is a periodic function with n period T, the mean function of Vt ( ) is periodic with period T. The autocorrelation function of Vt ( ) is given by R t t E V t V t * V ( + τ, ) = [ ( ) ( + τ )] = E[ an am] gt( t nt) gt( t+ τ mt). (8..3) n= m= Assume that the information sequence { a n } is wide-sense stationary with the autocorrelation function given by R ( n) = E[ a a ]. (8..4) a m n+ m
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 7-1 st Semester, 008 Then, (8..3) becomes R ( t+ τ, t) = R ( m n) g ( t nt) g ( t+ τ mt) V a T T n= m= = R ( m) g ( t nt) g ( t+ τ nt mt) m=. (8..5) a T T n= where the second summation gt( t nt) gt( t+ τ nt mt) is periodic with period T. n= Consequently, the autocorrelation function R ( t+ τ, t) is periodic in the variable t, that is, V R ( t+ T + τ, t+ T) = R ( t+ τ, t). (8..7) V V Since the random process Vt ( ) has a periodic mean function and a periodic autocorrelation function, the random process Vt ( ) is cyclostationary (see Definition 4..7).
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 8-1 st Semester, 008 The power-spectral density of a cyclostationary process Vt ( ) is determined by averaging the autocorrelation function R ( t+ τ, t) over a period T and then computing the Fourier transform of the V average autocorrelation function (see Corollary to Theorem 4.3.1). The average autocorrelation function of Vt ( ) is given by 1 R R t t dt T V( τ) = T T V( + τ, ) 1 = R m g t nt g t+ nt mt dt m= T a( ) T T( ) T( τ ) n T = 1 = R m g t g t+ mt dt m= T nt + a( ) T T( ) T( τ ) nt n T = 1 = Ra( m) gt( t) gt( t+ τ mt) dt T (8..8) m= The integral in (8..8) is the time-autocorrelation function of gt () t which is defined as (See (.3.1)) R ( τ ) g ( t) g ( t+ τ ) dt. (8..9) g T T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 9-1 st Semester, 008 From (8..8) and (8..9) it becomes 1 RV( τ) = Ra( m) Rg( τ mt) (8..10) T m= which has the form of a convolution sum. Hence the Fourier transform of the average autocorrelation function of Vt ( ) in (8..10) is given by jπ fτ SV( f) = RV( τ ) e d where ( ) a τ 1 = T jπ fτ Ra( m) Rg( τ mt) e d m= τ 1 T S f is the power spectral density of the information sequence { a } given by = Sa( f) GT( f) (8..11) n f mt Sa( f) Ra( m) e π m= = (8..1) and G ( f ) is the transfer function of the transmitting filter. T ( ) GT f is the Fourier transform of g ( ) R τ.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 30-1 st Semester, 008 From (8..11), notice that the power-spectral density SV ( f ) of the transmitted signal Vt ( ) depends on (1) the transfer function GT ( f ) of the transmitting filter and () the power spectral density Sa( f ) of the information sequence { a n }. Both GT ( f ) and Sa( f ) can be designed to adjuct the shape of the power spectral density of the transmitted signal. Examine the dependence of SV ( f ) on Sa( f ). First, we observe that, for an arbitrary autocorrelation function Ra ( m ), the corresponding power-spectral density S ( f ) is periodic in frequency with period a 1 T. Note that Sa( f ) in (8..1) is an exponential Fourier series with the Fourier coefficients { Ra ( m )}.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 31-1 st Semester, 008 Consequently, the autocorrelation function of the information sequence { a n } is given by 1 T ( ) 1 ( ) j π f mt a a T R m = T S f e df. (8..13) Second, suppose that the information symbols in the sequence { a n } are mutually uncorrelated. Then, the autocorrelation function of the information sequence { a n } is given by σ a + ma, m= 0, Ra ( m) = ma, m 0, (8..14) where σ = E[ a ] m is the variance of an information symbol. a n a From (8..1) and (8..14), the power-spectral density of the information sequence { a n } is given by jπ fmt a( ) σ a a m= S f = + m e. (8..15) The second term on the RHS of (8..15) is periodic with period 1 T and can be viewed as the Fourier series of a periodic sequence of impulses each of which has an area 1 T (see Table.1).
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 3-1 st Semester, 008 Therefore, (8..15) is expressed as m S ( f) = + m δ( f ). (8..16) T 1 a σa a T m= From (8..11) and (8..16), the power-spectral density of the transmitted signal Vt, ( ) when the sequence of information symbols is uncorrelated, is given by σ a ma m m SV( f) = GT( f) + G T δ f. (8..17) T T m= T T The first term on the RHS of (8..17), GT ( f ). σ T a G ( f), is a continuous dunction and its shape depends of T The second term in (8..17) consists of discrete frequency components spaced 1 T apart in frequency. Each component (or spectral line) has power that is proportional to ( ) GT f at f m =. T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 33-1 st Semester, 008 Note that the discrete frequency components can be eliminated by selecting the information symbol sequence { a n } to have zero mean. The mean m a in digital PAM, PSK, or QAM signals is easily forced to be zero by selecting the signal constellation points to be symmetrically positioned in the complex plane relative to the origin. Under the condition that m a = 0, the power-spectral density of the transmitted signal Vt ( ) becomes σ SV( f) G ( f) T a = T. (8..18) Thus, the system designer can control the spectral characteristics of the transmitted digital PAM signal. Ex. 8..1 Determine the power-spectral density of the transmitted signal Vt ( ) in (8..17), when g () t is the rectangular pulse shown in Figure 8.5(a). T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 34-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 35-1 st Semester, 008 Solution The Fourier transform of g () t is given by sinπ ft jπ ft GT ( f) = AT e. π ft T Hence, sinπ ft GT ( f) = ( AT) π ft = ( AT) sinc ( ft) which is shown in Figure 8.5(b). Note that it contains nulls at multiples of 1 T in frequency and that it decays inversely as the square of the frequency variable. As a consequence of the spectral nulls in GT ( f ), all but one of the discrete spectral components in (8..17) vanish.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 36-1 st Semester, 008 Thus, upon substitution for ( ) GT f into (8..17), we obtain sinπ ft SV( f) = σ aa T π ft = σ A Tsinc ( ft) + A m δ( f). a a Ex. 8.. Consider a binary sequence { b n }, from which we form the symbols an = bn + bn 1. The { b n } are assumed to be uncorrelated binary valued ( ± 1) random variables, each having zero mean and unit variance. Determine the power-spectral density of the transmitted signal.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 37-1 st Semester, 008 Solution The autocorrelation function of the sequence { a n } is given by Ra( m) = E[ ana n + m] = E[( b + b )( b + b )] n n 1 n+ m n+ m 1, m = 0, = 1, m =± 1, 0, otherwise. Hence, the power-spectral density of the input sequence is given by S ( f) = (1 + cos π ft) a = 4cos π ft and, from (8..17), the corresponding power spectral density for the modulated signal is given by 4 ( ) = ( ) cos π. T SV f GT f ft Figure 8.6 shows the power-density spectrum Sa( f ) of the input sequence, and the corresponding SV ( f ) when G ( f ) is the spectrum of the rectangular pulse. T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 38-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 39-1 st Semester, 008 As demonstrated in the example, the power spectral density of the transmitted signal can be shaped by having a correlated sequence { a n } as the input to the modulator. 8.. Power Spectral Density of a Carrier-Modulated Signal The autocorrelation function of the information sequence { a n } is given by R m = E a a +. (8..1) * a( ) [ n n m] The power spectral density of the information sequence { a n } is given by jπ fmt = a. (8..0) m= S ( f) R ( m) e a In Section 8..1, it was shown that the power spectral density of the equivalent baseband signal vt ( ) in (8..1) for bandpass PAM, QAM, and PSK is given by
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 40-1 st Semester, 008 1 S f S f G f V( ) = a( ) T( ). (8..19) To find out the relationship between the power spectral density of the baseband signal to the power spectral density of the bandpass signal, consider the bandpass PAM signal as an example. The autocorrelation function of the bandpass signal ut () = vt ()cosπ ft c is given by R (, ) [ ( ) ( )] U t+ τ t = E U t+ τ U t = E[ VV ( t)]cos π f ( t + τ) cosπ f t c = R ( t+ τ, t)cos π f ( t+ τ) cosπ f t V c c 1 = R ( t+ τ, t)[cosπ fτ + cos π f ( t+ τ)]. V c c c Then, the average of R ( t+ τ, t) over a single period T is given by U 1 RU( τ ) = RV( τ)cosπ fcτ, (8..)
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 41-1 st Semester, 008 as the second term in the previous equation involving the double frequency averages to zero for each period of cos4π fct. The power spectral density of the bandpass signal ut ( ) is given by jπ fτ SU( f) = RU( τ ) e d τ 1 = [ S ( f f ) + S ( f + f )]. (8..3) 4 V c V c Although (8..3) was derived for PAM, it also applies to QAM and PSK. The bandpass PAM, QAM, and PSK signals differ only in the autocorrelation function Ra ( m ) of the sequence { a n } and, hence, in the power spectral density Sa( f ) of { a n }.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 4-1 st Semester, 008 8.3 Signal Design for Bandlimited Channels Consider the problem of designing a bandlimited transmitting filter under thae condition that there is no channel distortion. Since H ( f) = C( f) G ( f), for distortion-free transmission the frequency response characteristic of the T channel has a constant magnitude and a linear phase over the bandwidth of the transmitted signal, that is, jπ f t0 Ce 0, f W, C( f) = 0, f > W, (8.3.3) where W is the available channel bandwidth, t 0 represents an arbitrary finite delay, which we set to zero for convenience, and C 0 is a constant gain factor which we set to unity for convenience. Thus, the distortion-free channel, H ( f) = G ( f) for f W and zero for f > W. T Consequently, the matched filter has a frequency response H ( f) = G ( f) and its output at the periodic * * T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 43-1 st Semester, 008 sampling times t = mt has the form ymt ( ) = x(0) am + axmt n ( nt) + vmt ( ) (8.3.4) or, simply n= n m y = x a + a x + v (8.3.5) m 0 m n m n m n= n m where x() t = g () t g () t and T R vt () is the output response of the matched filter to the input AWGN process nt ( ). The second term on the RHS of (8.3.5) represents the ISI. The amount of ISI and noise in the received signal can be viewed on the vertical input of an oscilloscope. The received signal is displayed on the vertical input with the horizontal sweep rate set at 1 T,. The resulting oscilloscope display is called an eye pattern because of its resemblance to the human eye.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 44-1 st Semester, 008 Two examples of eye patterns for binary PAM and quaternary (or 4-ary, M = 4) PAM are shown in Figure 8.7(a). The effect of ISI is to cause the eye to close, thereby reducing the margin for additive noise to cause errors which is shown in Figure 8.7(b). Note that ISI distorts the position of the zero crossings and causes a reduction in the eye opening. As a consequence, the system is more sensitive to a synchronization error and has a smaller margin against additive noise.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 45-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 46-1 st Semester, 008 8.3.1 Design of Bandlimited Signals for Zero ISI-The Nyquist Criterion In a digital communication system over a bandlimited channel, the Fourier transform of the signal at the output of the receiving filter is given by X( f) = G ( f) C( f) G ( f) T R where GT ( f ) and GR( f ) denote the transmitter and receiver filters frequency response, respectively, and C( f ) denotes the frequency response of the channel. We have also seen that the output of the receiving filter, sampled at t = mt, is given by ym = x(0) am + x( mt nt) an + v( mt). (8.3.6) n= n m To remove the effect of ISI, it is necessary and sufficient that x( mt nt ) = 0 for n m and x(0) 0, where we can assume x (0) = 1, without loss of generality. The choice of (0) x is equivalent to the choice of a constant gain factor in the receiving filter. This constant gain factor has no effect on the overall system performance since it scales both the signal and the noise.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 47-1 st Semester, 008 This implies that the overall communication system has to be designed such that 1, n = 0, xnt ( ) = 0, n 0. (8.3.7) In this section, we derive the necessary and sufficient condition for X( f ) in order for x( t ) to satisfy the above relation which is known as the Nyquist pulse-shaping criterion or Nyquist condition for zero ISI. Theorem 8.3.1 [Nyquist] A necessary and sufficient condition for x( t ) to satisfy 1, n = 0, xnt ( ) = 0, n 0, (8.3.8) is that its Fourier transform X( f ) satisfy m X f + = T T. (8.3.9) m= Proof In general, x( t ) is the inverse Fourier transform of X( f ).
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 48-1 st Semester, 008 Hence, x() t X( f) e j π f t df =. (8.3.10) At the sampling instants t x( nt ) X ( f ) e j π f nt df = nt, this relation becomes =. (8.3.11) Break up the integral in (8.3.11) into integrals covering the finite range of 1 T. Then, we obtain m+ 1 T jπ f nt x( nt ) = m 1 X ( f ) e df m= T m = + m= T 1 T 1 T jπ f nt X f e dt 1 T m = 1 + T m= T jπ f nt X f e dt 1 T 1 ( ) j π = Z f e f nt dt (8.3.1) T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 49-1 st Semester, 008 where m Z( f) = X f + m= T. (8.3.13) As Z( f ) is a periodic function with period 1 T coefficients { z n } as, it can be expanded in terms of its Fourier series j nft zne π m= Z( f) = (8.3.14) where 1 T jπ nft zn T 1 Z( f) e df T =. (8.3.15) Comparing (8.3.15) and (8.3.1) we obtain z = T x( nt). (8.3.16) n
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 50-1 st Semester, 008 Therefore, the necessary and sufficient conditions for (8.3.8) to be satisfied is that z n T, n= 0, = 0, n 0, (8.3.17) which, when substituted into (8.3.14), yields Z( f) or, equivalently, m= = T (8.3.18) m X f + = T T (8.3.19) which concludes the proof of the theorem. Now, suppose that the channel has a bandwidth of W. Then, C( f) 0 for f > W and consequently, X( f ) = 0 for f > W. ( X( f) = G ( f) C( f) G ( f) ) T R We distinguish three cases:
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 51-1 st Semester, 008 1. T < 1 W, or equivalently, 1 W T >. + n Since Z( f) = X f + n= T consists of nonoverlapping replicas of X ( f ), separated by 1 T as shown in Figure 8.8, there is no choice for X( f ) to achieve Z( f) T no ISI., and there is no way to design a system with
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 5-1 st Semester, 008. 1 T = W, or equivalently, 1 T The replicas of X( f ), separated by = W (the Nyquist rate). 1 T, are about to overlap as shown in Figure 8.9. It is clear that there exists only one X( f ) that results in Z( f) = T, that is,
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 53-1 st Semester, 008 T, f < W, X( f) = 0, otherwise, (8.3.0) f or, X( f) = T Π, which results in W xt () sinc t = T. (8.3.1) This impliess that the smallest value of T for which transmission with zero ISI possible is x() t has to be a sinc function. T = 1 W and The difficulty with this choice of x( t ) is that it is non-causal and therefore non-realizable. 0 To make it realizable, usually a delayed version of it, that is, sinc t t T is used and t 0 is chosen large t t0 enough such that for t < 0, we have sinc 0. T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 54-1 st Semester, 008 With this choice of x( t ), the sampling time must also be shifted to mt + t0. 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, consequently, a small mistiming error in sampling the output of the matched t 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. 3. For 8.10. T > 1 W, Z ( f ) consists of overlapping replications of X( f ) separated by 1 T, as shown in Figure
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 55-1 st Semester, 008 There exist numerous choices for X( f ), such that Z( f) T. For the T > 1 W case, a particular pulse spectrum that has been widely used is the raised cosine spectrum of which frequency characteristic is given as (see Problem 8.11)
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 56-1 st Semester, 008 1 α T, 0 f, T 1 α 1+ α X ( ) 1 rc f = T πt α f, 1+ cos f, T T α T 1+ α 0, f >, T (8.3.) where α is called the rolloff factor, 0 α 1. The bandwidth occupied by the signal beyond the Nyquist frequency and is usually expressed as a percentage of the Nyquist frequency. 1 T is called the excess bandwidth For example, when 1 α =, the excess bandwidth is 50%, and when α = 1 the excess bandwidth is 100%. The pulse having the raised cosine spectrum is given by πt πα t sin cos xt () = T T πt 4α t 1 T T
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 57-1 st Semester, 008 πα t t cos = sinc( ) T T 4α t 1 T. (8.3.3) Note that x( t ) is normalized so that x (0) = 1. Figure 8.11 shows the raised cosine spectral characteristics and the corresponding pulse for 1 α = 0,,1.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 58-1 st Semester, 008 Figure 8.11 Pulses having a raised cosine spectrum.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 59-1 st Semester, 008 Note that for α = 0, the pulse reduces to xt () sinc t = T, and the symbol rate 1 W T =. When α = 1, the symbol rate is 1 W T =. 1 In general, the tails of x( t ) decay as t 3 for α > 0. Consequently, a mistiming error in sampling leads to a series of intersymbol interference components that converges to a finite value. Thanks to smooth characteristics of the raised cosine spectrum, it is possible to design practical filters for the transmitter and receiver that approximate the overall desired frequency responseing filter In this case, if the receiver filter is matched to the transmitting filter, we have X ( f) = G ( f) G ( f) rc T R = GT ( f).
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 60-1 st Semester, 008 Ideally, G ( f) X ( f) e T jπ ft0 = (8.3.5) rc and G f = G f, * R( ) T( ) where t 0 is some nominal delay that is required to assure physical realizability of the filter. Thus, the overall raised cosine spectral characteristic is split evenly between the transmitting filter and the receiving filter. Also note that an additional delay is necessary to ensure the physical realizability of the receiving filter.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 61-1 st Semester, 008 8.3. Design of Bandlimited Signals with Controlled ISI-Partial Response Signals To achieve zero ISI, it is necessary to reduce the symbol rate 1 T below the Nyquist rate of W symbols/sec in order to realize practical transmitting and receiving filters. By relaxing the condition of zero ISI that x( nt ) = 0 for n 0, we can achieve a symbol transmission rate of W symbols/sec. Suppose that we design the bandlimited signal to have controlled ISI at one time instant which implies that we allow one additional nonzero value in the samples { x( nt )}. The ISI that we introduce is deterministic or controlled and, hence, it can be taken into account at the receiver.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 6-1 st Semester, 008 One special case that leads to (approximately) physically realizable transmitting and receiving filters is specified by the samples 1, n = 0, xnt ( ) = 1, n= 1, 0, otherwise. (8.3.6) By using (8.3.16), we obtain T, n= 1, zn = T, n= 0, 0, otherwise (8.3.7) From (8.3.14) and (8.3.7) we have Z( f) T Te jπ ft = + (8.3.8) It is impossible to satisfy (8.3.8) for T < 1 W. It is convenient to deal with samples of x() t that are normalized to unity for n = 0,1.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 63-1 st Semester, 008 However, for 1 T = W, we obtain 1 1 j π f + e W, f < W, X( f) = W 0, otherwise, 1 = W π f j π f W e cos, W f < W, 0, otherwise. (8.3.9) Therefore, x( t ) is given by x() t = sinc( Wt) + sinc(wt 1) (8.3.30) which is called a duobinary signal pulse shown in Figure 8.1.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 64-1 st Semester, 008
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 65-1 st Semester, 008 Note that the spectraum decays to zero smoothly, which means that physically realizable filters can be designed that approximate this spectrum very closely. Thus, a symbol rate of W is achieved. Another special case that leads to (approximately) physically realizable transmitting and receiving filters is specified by the samples n x = xnt ( ) W 1, n = 1, = 1, n = 1, 0, otherwise. (8.3.31) The corresponding pulse x( t ) is given by t+ T t T xt ( ) = sinc( ) sinc( ) (8.3.3) T T and its spectrum is given by
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 66-1 st Semester, 008 π f π f 1 j j W W e e, f W, X( f) = W 0, f > W, j π f sin, f W, = W W 0, f > W, (8.3.33) which is shown in Figure 8.13. It is called a modified duobinary signal pulse. Note that the spectrum of this signal has a zero at f = 0, making it suitable for transmission over a channel that does not pass D.C.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 67-1 st Semester, 008 Figure 8.13 Time domain and frequency domain characteristics of a modified duobinary signal.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 68-1 st Semester, 008 One can obtain other interesting and physically realizable filter characteristics by selecting different values n for the samples x( ) W and more than two nonzero samples. However, as we select more nonzero samples, solving the controlled ISI becomes more cumbersome and impractical. In general, the class of bandlimited signals pulses that have the form n sin πwt ( ) n xt () = ( ) W x (8.3.34) n W n = πwt ( ) W and their corresponding spectra 1 n jnπ f x( ) e, f W, X( f) = W n= W (8.3.35) 0, f > W, are called partial response signals when controlled ISI is ISI is purposely introduced by selecting two or n more nonzero samples from the set x( ) W.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 69-1 st Semester, 008 The resulting signal pulses allow us to transmit information symbols at the Nyquist rate of W symbols per second. 8.4 Probability of Error in Detection of PAM We evaluate the performance of the receiver for demodulating and detecting an M -ary PAM signal in the presence of additive white Gaussian noise at its input for two cases: 1) the case that the transmitting and receiving filters GT ( f ) and GR( f ) are designed for zero ISI and ) the case that GT ( f ) and GR( f ) are designed such that x() t = gt() t gr() t is either a duobinary signal or a modified duobinary signal. The results can be generalized to two-dimensional modulations, such as PSK and QAM, and multidimensional signals.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 70-1 st Semester, 008 8.4.1 Probability of Error for Detection of PAM with Zero ISI In the absence of ISI, the received signal sample at the output of the receiving matched filter is given by y = x a + v (8.4.1) where m 0 m m W x0 = GT ( f) df W = ε g (8.4.) and v m is the additive Gaussian noise which has zero mean and variance ε N g 0 σ v =. (8.4.3) Suppose that a m takes one of M possible equally spaced amplitude values with equal probability. Then, the probability of error for digital PAM in a bandlimited additive white Gaussian noise channel in the absence of ISI is identical to that for M -ary PAM as given in Section 7.6..
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 71-1 st Semester, 008 The probability of error for digital PAM is given by P M ( M 1) ε g = Q. (8.4.4) M N 0 3ε av Since ε g = M 1, where ε av = kεbav is the average energy per symbol and ε bav is the average energy/bit, (8.4.4) becomes P M ( M 1) 6(log M) ε bav = Q M ( M 1) N 0 (8.4.5) which is exactly the form for the probability of error of M -ary PAM derived in Section 7.6.. While there was no bandwidth constraint on the PAM signals in Section 7.6., here the transmitted signal pulses are designed to be bandlimited and to have zero ISI. Nevertheless, the receivers (demodulators and detectors) in both cases are optimum (matched filters) for the corresponding transmitted signals. Consequently, no loss in error-rate performance results from the bandwidth constraint when the signal pulse is designed for zero ISI and the channel does not distort the transmitted signal.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 7-1 st Semester, 008 8.4. Symbol-by-Symbol Detection of Data with Controlled ISI Consider a symbol-by-symbol method for detecting the information symbols at the demodulator when the received signal contains controlled ISI. In particular, we consider the detection of the duobinary and the modified duobinary partial response signals. In both cases, assume that the desired spectral characteristic X( f ) for the partial response signal is split evenly between the transmitting and receiving filters, that is, G ( f) = G ( f) = v X( f). T R 1 For the duobinary signal pulse, 1, n = 0, xnt ( ) = 1, n= 1, 0, otherwise. Hence, the samples at the output of the receiving filter are given by ym = bm + vm = a + a + v (8.4.6) m m 1 m
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 73-1 st Semester, 008 where { a m } is the transmitted sequence of amplitudes and { v m } is a sequence of additive Gaussian noise samples. Ignore the noise for the moment and consider the binary case where a m = ± 1 with equal probability. Then, b takes on one of three possible values, namely, b =, 0, with corresponding probabilities m m 1 1 1,, 4 4. If the detected symbol from the ( m 1) st signaling interval is a 1, its effect on the received signal in the m m th signaling interval b m can be eliminated by subtraction, thus allowing a m to be detected. This process can be repeated sequentially for every received symbol. The major problem with this procedure is that errors arising from the additive noise tend to propagate. That is, if am 1 is in error, its effect on b m is not eliminated but, in fact, it is reinforced by the incorrect
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 74-1 st Semester, 008 subtraction so that the detection of a m is also likely to be in error. Error propagation can be avoided by precoding the data on the binary data sequence prior to modulation at the transmitter instead of eliminating the controlled ISI by subtraction at the receiver. From the data sequence { d n } of 1 s and 0 s that is to be transmitted, a new sequence { p n }, called the precoded sequence, is generated. For the duobinary signal, the precoded sequence is given by p m = d pm 1, m= 1,,, (8.4.7) m where denotes modulo- subtraction. Then, set a m = 1 if p m = 0 and a m = 1 if p m = 1, that is, am = pm 1. The noise-free samples at the output of the receiving filter are given by Although this is identical to modulo- addition, it is convenient to view the precoding operation for duobinary in terms of modulo- subtraction.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 75-1 st Semester, 008 b = a + a m m m 1 = (p 1) + ( 1) m pm 1 = ( p + 1). (8.4.8) m pm 1 Consequently, it becomes p m bm + pm 1 = + 1. (8.4.9) Since dm = pm pm 1, it follows that bm d m = + 1(mod ). (8.4.10) Consequently, if b m = ±, d m = 0 and if b m = 0, d m = 1. An example that illustrates the precoding and decoding operations is given in Table 8.1.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 76-1 st Semester, 008 Table 8.1 Binary Signaling with Duobinary Pulses. In the presence of additive noise the sampled outputs from the receiving filter are given by (8.4.6). In this case ym = bm + vm is compared with the two thresholds set at + 1 and 1. The data sequence { d n } is obtained according to the detection rule
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 77-1 st Semester, 008 d m 1, 1 < ym < 1, = 0, ym 1. (8.4.11) The extension from binary PAM to multilevel PAM signaling using the duobinary pulse is straightforward. In this case the M -level amplitude sequence { a m } results in a (noise-free) sequence bm = am + am 1, m = 1,,, (8.4.1) which has M 1 possible equally spaced levels. The amplitude levels are determined from the relation a = p ( M 1) (8.4.13) m m where { p m } is the precoded sequence that is obtained from an M -level data sequence { d m } according to the relation (see (8.4.7)) p m = dm p (mod ) m 1 M (8.4.14) where denotes modulo- M subtraction and the possible values of the sequence { d m } are 0,1,,, M 1.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 78-1 st Semester, 008 In the absence of noise, the samples at the output of the receiving filter are expressed as b = a + a m m m 1 = [ p + p ( M 1)]. (8.4.15) m m 1 Hence, bm pm + pm 1 = + ( M 1). (8.4.16) Since d = p + p (mod ) 1 M, it follows that m m m d m bm = + ( M 1) (mod M ). (8.4.17) An example illustrating multilevel precoding and decoding is given in Table 8..
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 79-1 st Semester, 008 Table 8. Four-Level Transmission with Duobinary Pulses. In the presence of noise, the received signal-plus-noise is quantized to the nearest of the possible signal levels and the rule given above is used on the quantized values to recover the data sequence.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 80-1 st Semester, 008 In the case of the modified duobinary pulse, the controlled ISI is given by 1, n = 1, n x( ) = 1, n= 1, W 0, otherwise. Consequently, the noise-free sampled output from the receiving filter is given by bm am am = (8.4.18) where the M -level sequence { a n } is obtained by mapping a precoded sequence according to the relation (8..43) and p = d p (mod ) M. (8.4.19) m m m From these relations, it is shown that the detection rule for recovering the data sequence { d m } from { b m } in the absence of noise is given by bm dm = (mod M). (8.4.0) Note that the precoding of the data at the transmitter makes it possible to detect the received data on a symbol-by-symbol basis without having to look back at previously detected symbols.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 81-1 st Semester, 008 Thus, error propagation is avoided. The symbol-by-symbol detection rule described above is not the optimum detection scheme for partial response signals. Nevertheless, symbol-by-symbol detection is relatively simple to implement and is used in many practical applications involving duobinary and modified duobinary pulse signals.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 8-1 st Semester, 008 8.4.3 Probability of Error for Detection of Partial Response Signals Assume an ideal bandlimited channel with additive white Gaussian noise, The model for the communications system is shown in Figure 8.14.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 83-1 st Semester, 008 Symbol-by-Symbol Detector At the transmitter, the M -level data sequence { d n } is precoded as described previously. The precoder output is mapped into one of M possible amplitude levels. Then the transmitting filter with frequency response G ( f ) has an output vt () = ag n T( t nt). (8.4.1) n= T The partial-response function X( f ) is divided equally between the transmitting and receiving filters. Hence, the receiving filter is matched to the transmitted pulse, and the cascade of the two filters results in the frequency characteristic G ( f) G ( f) = X( f). (8.4.) T R The matched filter output is sampled at t = nt = and the samples are fed to the decoder. W
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 84-1 st Semester, 008 For the duobinary signal, the output of the matched filter at the sampling instant is given by y = a + a + v m m m 1 m = b + v (8.4.3) m m where v m is the additive noise component. Similarly, the output of the matched filter for the modified duobinary signal is given by y = a a + v m m m m = b + v. (8.4.4) m m For binary transmission, let am = ± d, where d is the distance between signal levels. Then, the corresponding values of b are ( d, 0, d). m For M -ary PAM signal transmission, where a = ± d, ± 3 d,, ± ( M 1) d, the received signal levels are given by b = 0, ± d ± 4 d,, ± ( M 1) d. m m
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 85-1 st Semester, 008 Hence, the number of received levels is M 1. Assume that the input transmitted symbol { a m } are equi-probable. Then, for duobinary and modified duobinary signals in the absence of noise, the received output levels have a (triangular) probability mass function of the form given by M j pb ( = jd) =, j = 0, ± 1, ±,, ± ( M 1), (8.4.5) M where b is the noise-free received level and d is the distance between any two adjacent received signal levels. The channel corrupts the transmitted signal by adding white Gaussian noise with zero-mean and powerspectral density. N 0 Assume that a symbol error is committed whenever the magnitude of the additive noise exceeds the distance d. This assumption neglects the rare event that a large noise component with magnitude exceeding d may result in a received signal level that yields a correct symbol decision.
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 86-1 st Semester, 008 The noise component σ W 0 v = R( ) W v m is Gaussian with zero-mean and variance given by N G f df N W 0 = X( f) df W N = 0. π (8.4.6) for both the duobinary and the modified duobinary signals. Hence, the symbol probability of error is upper-bounded as M P < P( y jd > d b= jd) P( b= jd) M j= ( M ) ( ) ( ) + P y+ ( M 1) d > d b= ( M 1) d P b= ( M 1) d M 1 = P( y > d b= 0) Pb ( = jd) Pb ( = 0) Pb= ( M 1) d j= 0 ( ) 1 = 1 P( y > d b= 0). (8.4.7) M
Chapter 8. Digital Transmission through Bandlimited AWGN Channels - 87-1 st Semester, 008 But x σ v P( y > d b= 0) = e dx d πσ v π d = Q. (8.4.8) N 0 Therefore, the average probability of symbol error is upper-bounded as P M 1 π d < 1 Q. (8.4.9) M N 0 For the M -ary PAM signal with equi-probable transmitted level, the average power at the output of the transmitting filter is given by Ea [ ] W m Pav GT ( f) df W = T Ea [ m] = T W W 4 E a m πt X( f) df = [ ] (8.4.30)