Tse and Viswanath:Fundamentals of Wireless Communication 35 [(t】 1 s(t)】 + -s(t) √2c0s2xf.t 1 S[() Figure 2.8:Illustration of upconversion from s(t)to s(t)and followed by downcon- version from s(t)back to s(t). but chosen to normalize the energies of s(t)and s(t)to be the same.Note that s(t) is bandlimited in [-W/2,W/2].See Figure 2.7. To reconstructs()from(),we observe that V2S(f)=S(f-f)+S8(-f-f): (2.22) Taking inverse Fourier transforms,we get s(t)= s(e(e-=v (2.23) In terms of real signals,the relationship between s(t)and s(t)is shown in Figure 2.8.The passband signal s(t)is obtained by modulating s(t)]by v2cos2rfet and 3[s6(t)]by -v2sin 2fet and summing,to gets(t)ei2e](up-conversion). nd sigmn)espectively)is otained by by V2cos2 ft (re 2ft)followed by ideal low-pass baseband [-W/2,W/2](down-co Let us now go back to the multipath fading channel (2.14)with impulse response given by (2.18).Let (t)and y(t)be the complex baseband equivalents of the trans- mitted signal x(t)and the received signal y(t),respectively.Figure 2.9 shows the system diagram from z(t)to (t).This implementation of a passband communica- tion system is known as quadrature amplitude modulation(QAM).The signal Rr(t) issometimes calld the in-phase,I,and the qudrature componen
Tse and Viswanath: Fundamentals of Wireless Communication 35 X X X X ✘ ✙✚ ✛ ✜✢ ✣ ✤ ✥ ✙✚ ✛ ✜✢ ✣ ✤ ✦ ✧ ★ ✩ ✪ ✧✫ ✬ ✭ ✢ ✦ ✧ ★ ✩ ✪ ✧ ✫ ✬ ✭ ✢ ✮ ✯ ✰ ✮ ✯ ✰ ✮ ✯ ✰ ✮ ✯ ✰ ✱ ✱ ✥ ✙✚ ✛ ✜✢ ✣ ✤ ✘ ✙✚ ✛ ✜✢ ✣ ✤ ✮ ✲ ✰ ✮ ✲ ✰ ✴ ✳ ✚ ✜✢ ✣ ✴ Figure 2.8: Illustration of upconversion from sb(t) to s(t) and followed by downconversion from s(t) back to sb(t). but chosen to normalize the energies of sb(t) and s(t) to be the same. Note that sb(t) is bandlimited in [−W/2, W/2]. See Figure 2.7. To reconstruct s(t) from sb(t), we observe that √ 2S(f) = Sb(f − fc) + S ∗ b (−f − fc). (2.22) Taking inverse Fourier transforms, we get s(t) = 1 √ 2 sb(t)e j2πfct + s ∗ b (t)e −j2πfct = √ 2< sb(t)e j2πfct . (2.23) In terms of real signals, the relationship between s(t) and sb(t) is shown in Figure 2.8. The passband signal s(t) is obtained by modulating <[sb(t)] by √ 2 cos 2πfct and =[sb(t)] by − √ 2 sin 2πfct and summing, to get < sb(t)e j2πfct (up-conversion). The baseband signal <[sb(t)] (respectively =[sb(t)]) is obtained by modulating s(t) by √ 2 cos 2πfct (respectively √ 2 sin 2πfct) followed by ideal low-pass filtering at the baseband [−W/2, W/2] (down-conversion). Let us now go back to the multipath fading channel (2.14) with impulse response given by (2.18). Let xb(t) and yb(t) be the complex baseband equivalents of the transmitted signal x(t) and the received signal y(t), respectively. Figure 2.9 shows the system diagram from xb(t) to yb(t). This implementation of a passband communication system is known as quadrature amplitude modulation (QAM). The signal <[xb(t)] is sometimes called the in-phase component, I, and =[xb(t)] the quadrature component,
Tse and Viswanath:Fundamentals of Wireless Communication 36 Q.(rotated by /2.)We now calculate the baseband equivalent channel.Substituting (t)=v(t)ei2fet]and y(t)=v(t)ej2rle]into (2.14)we get R%(d)c2网=∑a,(R[rt-T)c2-o] = R∑a,(),t-()e2=e (2.24) 1( Similarly,one can obtain (see Exercise 2.13) (t)]= ∑a,)t-(e2oe2f (2.25) Hence,the baseband equivalent channel is %国-∑e)t-r) (2.26) where a(t):=a(t)e-j2Jen() (2.27） The input-output relationship in (2.26)is also that of a linear time-varying system. and the baseband equivalent impulse response is ho(,t)=>a'(t)o(r-(t)). (2.28) This representation is easy to interpret in the time domain,where the effect of the carrier frequency can be seen explicitly.The baseband output is the sum,over each path,of the delayed replicas of the baseband input.The magnitude of the ith such term is the magnitude of the response on the given path;this changes slowly,with significant changes occurring on the order of seconds or more.The phase is changed byπ/2(i.e, is chan ged)when the delayon the p ath cha es by 1/(4fe) henth iscngngat veloctythe time required for sucp he path lengt ange oy a quarte length,i. by c/(4fe) is c/(4fev).Recalling that the Doppler shift D at frequency f is fu/c,and noting that ffe for narrow band communication,the time required for a /2 phase change is 1/(4D).For the single reflecting wall example,this is about 5 ms(assuming fe=900 MHz and v=60 km/h).The phases of both paths are rotating at this rate but in oppote that site directio f:t),i.e.,the frequency response of the original system (at a fixed t)shifted by the carrier frequency.This provides another way of thinking about the baseband equivalent channel
Tse and Viswanath: Fundamentals of Wireless Communication 36 Q, (rotated by π/2.) We now calculate the baseband equivalent channel. Substituting x(t) = √ 2<[xb(t)e j2πfct ] and y(t) = √ 2<[yb(t)e j2πfct ] into (2.14) we get < yb(t)e j2πfct = X i ai(t)< xb(t − τi(t))e j2πfc(t−τi (t)) , = < "(X i ai(t)xb(t − τi(t))e −j2πfcτi(t) ) e j2πfct # . (2.24) Similarly, one can obtain (see Exercise 2.13) = yb(t)e j2πfct = = "(X i ai(t)xb(t − τi(t))e −j2πfcτi(t) ) e j2πfct # . (2.25) Hence, the baseband equivalent channel is yb(t) = X i a b i (t)xb(t − τi(t)), (2.26) where a b i (t) := ai(t)e −j2πfcτi(t) . (2.27) The input-output relationship in (2.26) is also that of a linear time-varying system, and the baseband equivalent impulse response is hb(τ, t) = X i a b i (t)δ(τ − τi(t)). (2.28) This representation is easy to interpret in the time domain, where the effect of the carrier frequency can be seen explicitly. The baseband output is the sum, over each path, of the delayed replicas of the baseband input. The magnitude of the i th such term is the magnitude of the response on the given path; this changes slowly, with significant changes occurring on the order of seconds or more. The phase is changed by π/2 (i.e., is changed significantly) when the delay on the path changes by 1/(4fc), or equivalently, when the path length changes by a quarter wavelength, i.e., by c/(4fc). If the path length is changing at velocity v, the time required for such a phase change is c/(4fcv). Recalling that the Doppler shift D at frequency f is fv/c, and noting that f ≈ fc for narrow band communication, the time required for a π/2 phase change is 1/(4D). For the single reflecting wall example, this is about 5 ms (assuming fc = 900 MHz and v = 60 km/h). The phases of both paths are rotating at this rate but in opposite directions. Note that the Fourier transform Hb(f;t) of hb(τ, t) for a fixed t is simply H(f + fc;t), i.e., the frequency response of the original system (at a fixed t) shifted by the carrier frequency. This provides another way of thinking about the baseband equivalent channel.
Tse and Viswanath:Fundamentals of Wireless Communicatior 37 R6(t) 1 -(x) (纫 本9h(,)@ 2cos2mf 3r(t】 Figure 2.9:System diagram from the baseband transmitted signal(t)to the baseband received signal (t). 2.2.3 A Discrete Time Baseband Model The next step in creating a useful channel model is to convert the continuous time A se e that the in t waveform (t)is bandlimit equivalent is then limited to W/2 and can be represented as ()=∑xn小sinc(Wt-n, (2.29) where x[n]is given by o(n/W)and sinc(t)is defined as sinc(t):=sin(t) (2.30) 元t This representation follows from the sampling theorem,which says that any waveform bandlimited to W/2 can be expanded in terms of the orthogonal basis [sinc(Wt-n) with coefficients given by the samples (taken uniformly at integer multiples of 1/W). Using(2.26),the baseband output is given by y(t)=>r[n]a(t)sinc (Wt-W(t)-n). (2.31) The sampled outputs at multiples of 1/W,y[m]:=(m/W),are then given by m-∑∑xm∑a(m/W)sinc[m-n-t(m/W)W] (2.32
Tse and Viswanath: Fundamentals of Wireless Communication 37 X X X X −π 2 −π 2 1 1 −W 2 W 2 W 2 −W 2 √ 2 cos 2πfct √ 2 cos 2πfct ˜ + ˜ h(τ, t) x(t) y(t) =[yb(t)] <[yb(t)] =[xb(t)] <[xb(t)] Figure 2.9: System diagram from the baseband transmitted signal xb(t) to the baseband received signal yb(t). 2.2.3 A Discrete Time Baseband Model The next step in creating a useful channel model is to convert the continuous time channel to a discrete time channel. We take the usual approach of the sampling theorem. Assume that the input waveform x(t) is bandlimited to W. The baseband equivalent is then limited to W/2 and can be represented as xb(t) = X n x[n]sinc(Wt − n), (2.29) where x[n] is given by xb(n/W) and sinc(t) is defined as sinc(t) := sin(πt) πt . (2.30) This representation follows from the sampling theorem, which says that any waveform bandlimited to W/2 can be expanded in terms of the orthogonal basis {sinc(Wt−n)}n, with coefficients given by the samples (taken uniformly at integer multiples of 1/W). Using (2.26), the baseband output is given by yb(t) = X n x[n] X i a b i (t)sinc (Wt − Wτi(t) − n). (2.31) The sampled outputs at multiples of 1/W, y[m] := yb(m/W), are then given by y[m] = X n x[n] X i a b i (m/W)sinc[m − n − τi(m/W)W]. (2.32)
Tse and Viswanath:Fundamentals of Wireless Communication 38 The sampled output ym]can equivalently be thought as the projection of the waveform (t)onto the waveform Wsinc(Wt-m).Let e:=m-n.Then m-∑rm-4∑(m/w))sinee-r(m/w)w (2.33) By defining helml:=>a(m/W)sincle-n(m/W)wl, (2.34) (2.33)can be written in the simple form gm=∑∑hem]rm-. (2.35) We denote helml as the cth(complex)channel filter tap at time m.Its value is a function of mainly the gains a(t)of the paths,whose delays T(t)are close to e/W path iant. wh the gains)'s and the delays)'s of the he=∑a'sincle-,W], (2.36) and the channel is linear time-invariant.The thtap can be interpreted as samples of the low -pass filtered baseband channel response)(c.f.(2.19) he=(h sinc)(e/W) (2.37) where is the convolution operation. We can interpret the sampling operation as modulation and a communication s system.At ti are modulating the complex symbol n](in phase plus quad rature comp onents) the sinc pulse re ef n.At the receiver,the received signal is sampled at timesm/W at the ouput of the ow-pass filter.Figure 2.11 shows the complete system.In practice,other transmit pulses,such as the raised cosine pulse,are often used in place of the sinc pulse,which has rather poor time-decay property and tends to be more susceptible to timing errors.This necessitates sampling at a rate below the Nyquist sampling rate,but does not alter the esse ential ture of the following descriptions.Hence we will onfe to Nyquis sampling. Due to the Doppler spread,the bandwidth of the outputy(t)is generally slightly larger than the bandwidth W/2 of the input o(t),and thus the output samples fy[m] do not fully represent the output waveform.This problem is usually ignored in practice
Tse and Viswanath: Fundamentals of Wireless Communication 38 The sampled output y[m] can equivalently be thought as the projection of the waveform yb(t) onto the waveform Wsinc(Wt − m). Let ` := m − n. Then y[m] = X ` x[m − `] X i a b i (m/W)sinc[` − τi(m/W)W]. (2.33) By defining h` [m] := X i a b i (m/W)sinc[` − τi(m/W)W], (2.34) (2.33) can be written in the simple form y[m] = X ` h` [m] x[m − `]. (2.35) We denote h` [m] as the ` th (complex) channel filter tap at time m. Its value is a function of mainly the gains a b i (t) of the paths, whose delays τi(t) are close to `/W (Figure 2.10). In the special case where the gains a b i (t)’s and the delays τi(t)’s of the paths are time-invariant, (2.34) simplifies to: h` = X i a b i sinc[` − τiW], (2.36) and the channel is linear time-invariant. The ` th tap can be interpreted as samples of the low-pass filtered baseband channel response hb(τ ) (c.f. (2.19)): h` = (hb ∗ sinc)(`/W). (2.37) where ∗ is the convolution operation. We can interpret the sampling operation as modulation and demodulation in a communication system. At time n, we are modulating the complex symbol x[n] (inphase plus quadrature components) by the sinc pulse before the up-conversion. At the receiver, the received signal is sampled at times m/W at the output of the low-pass filter. Figure 2.11 shows the complete system. In practice, other transmit pulses, such as the raised cosine pulse, are often used in place of the sinc pulse, which has rather poor time-decay property and tends to be more susceptible to timing errors. This necessitates sampling at a rate below the Nyquist sampling rate, but does not alter the essential nature of the following descriptions. Hence we will confine to Nyquist sampling. Due to the Doppler spread, the bandwidth of the output yb(t) is generally slightly larger than the bandwidth W/2 of the input xb(t), and thus the output samples {y[m]} do not fully represent the output waveform. This problem is usually ignored in practice
Tse and Viswanath:Fundamentals of Wireless Communication Main contribution-=0 Main contribution-1=0 Main contribution-I=1 Main contribution-I=2 Main contribution-1=2 ■0■11■2 Figure 2.10:Due to the decay of the sinc function,theth path contributes most significantly to the eth tap if its delay falls in the window [e/W-1/(2W),e/W+ 1/(2W)1
Tse and Viswanath: Fundamentals of Wireless Communication 39 1 W l = 0l = 1l = 2 Main contribution - l = 0 Main contribution - l = 1 Main contribution - l = 0 Main contribution - l = 2 Main contribution - l = 2 i = 0 i = 2 i = 1 i = 3 i = 4 Figure 2.10: Due to the decay of the sinc function, the i th path contributes most significantly to the ` th tap if its delay falls in the window [`/W − 1/(2W), `/W + 1/(2W)]