28. Narrowband Noise Representation

Transcription

1 Narrowband Noise Represenaion on Mac 8. Narrowband Noise Represenaion In mos communicaion sysems, we are ofen dealing wih band-pass filering of signals. Wideband noise will be shaped ino bandlimied noise. If he bandwidh of he bandlimied noise is relaively small compared o he carrier frequency, we refer o his as narrowband noise. Figure 8. shows how o generae narrowband noise from wideband noise. Figure 8. Generaion of narrowband noise. We can derive he power specral densiy G n (f) and he auo-correlaion funcion R nn (τ) of he narrowband noise and use hem o analyse he performance of linear sysems. In pracice, we ofen deal wih mixing (muliplicaion), which is a non-linear operaion, and he sysem analysis becomes difficul. In such a case, i is useful o express he narrowband noise as n() = x() cos πf c - y() sin πf c (8.) where f c is he carrier frequency wihin he band occupied by he noise. x() and y() are known as he quadraure componens of he noise n(). The Hiber ransform of n() is n^ () = H[n()] = x() sin πf c + y() cos πf c (8.) Proof. The Fourier ransform of n() is N(f) = X(f - f c ) + X(f+ f c ) + jy(f- f c ) - jy(f+ f c ) Le N^ (f) be he Fourier ransform of n^ (). In he frequency domain, N^ (f) = N(f)[-j sgn(f)]. We simply muliply all posiive frequency componens of N(f) by -j and all negaive frequency componens of N(f) by j. Thus, N^ (f) = -j X(f-f c )+ j X(f+ f c ) - j jy(f- f c ) - j jy(f+ f c ) N^ (f) = -j X(f - f c ) + j X(f+ f c ) + Y(f- f c ) + Y(f+ f c ) and he inverse Fourier ransform of N^ (f) is n^ () = x() sin πf c + y() cos πf c Q.E.D. 8.

2 Narrowband Noise Represenaion on Mac The quadraure componens x() and y() can now be derived from equaions (8.) and (8.). x() = n()cos πf c + n^ ()sin πf c (8.3) and y() = n()cos πf c - n^ ()sin πf c (8.4) Given n(), he quadraure componens x() and y() can be obained by using he arrangemen shown in Figure 8.. Figure 8. Generaion of quadraure componens of n(). x() and y() have he following properies:. E[x() y()] =. x() and y() are uncorrelaed wih each oher.. x() and y() have he same means and variances as n(). 3. If n() is Gaussian, hen x() and y() are also Gaussian. 4. x() and y() have idenical power specral densiies, relaed o he power specral densiy of n() by G x (f) = G y (f) = G n (f- f c ) + G n (f+ f c ) (8.5) for f c -.5B < f < f c +.5B and B is he bandwidh of n(). Proof. (Under Consrucion) Equaion (8.5) is he key ha will enable us o calculae he effec of noise on AM and FM sysems. I implies ha he power specral densiy of x() and y() can be found by shifing he posiive porion and negaive porion of G n (f) o zero frequency and adding o give G x (f) and G y (f). This is shown in Figure 8.3. Figure 8.3 (a) Power specral densiy of n(), (b) Power specral densiy of x() and y(). In he special case where G n (f) is symmerical abou he carrier frequency f c, he posiive- and negaive-frequency conribuions are shifed o zero frequency and added o give 8.

3 Narrowband Noise Represenaion on Mac G x (f) = G y (f) = G n (f- f c ) = G n (f+ f c ) (8.6) Example 8. Given ha he power specral densiy of a narrowband Gaussian noise of variance σ and power N is G n (f) = N (f f c ) πσ e σ + N (f +f c ) πσ e σ where f c is he carrier frequency wihin he band occupied by he noise, hen he power specral densiies of he quadraure componens of he noise are G x (f) = G y (f) = G n (f+ f c ) = N This is shown in Figure 8.4. πσ e f σ Figure 8.4 (a) Power specral densiy of narrowband Gaussian noise n(), (b) Power specral densiy of x() and y(). Performance of Binary FSK Figure 8.5 Synchronous deecion of binary FSK signals. Consider he synchronous deecor of binary FSK signals shown in Figure 8.5. In he presence of addiive whie Gaussian noise (AWGN), w(), he received signal is r() = Acos πf c + w() where A is a consan and f c is he carrier frequency employed if a has been sen. The signals a he oupu of he band-pass filers of cenre frequencies f c and f c are r () = Acos πf c + n () and 8.3

4 Narrowband Noise Represenaion on Mac r () = n () where n () = x () cos πf c - y () sin πf c and n () = x () cos πf c - y () sin πf c are he narrowband noise. Wih appropriae design of low-pass filer and sampling period, he sampled oupu signals are v o = A + x v o = x and v = A + [x - x ]. x and x are saisically independen Gaussian random variables wih zero mean and fixed variance σ = N, where N is he power of he random variable. I can be seen ha one of he deecors has signal plus noise, he oher deecor has noise only. When f c is he carrier frequency employed for sending a, he received signal is r() = Acos πf c + w(). I can be shown ha v = -A + [x - x ] Since E [x - x ] = E [x ] - E [x x ] + E [x ] = E [x ] + E [x ] = σ + σ, he oal variance σ = σ. The wo disribuions of v Figure 8.6. are shown in Figure 8.6 Condiional probabiliy densiy funcion. 8.4

5 The condiional probabiliy densiy funcion of v assuming a is sen is Narrowband Noise Represenaion on Mac f(v/) = πσ v A ( + ) e σ and he probabiliy of error given a is sen is P e = f(v/)dv ( v A + ) = e σ dv πσ (8.7) Similarly, he probabiliy of error given a is sen is P e = πσ ( v A ) e σ dv = P e Le p be he probabiliy of sending a and p be he probabiliy of sending a. For equally likely ransmission of binary signals, we have p = p =.5. The average probabiliy of error is given by P e = p P e + p P e = P e Le u = v + A σ. Then u = ( v + A) σ equaion (8.7), we ge and du = dv σ. Subsiuing u and du ino P e = u = A σ π e -u du = [ π e-u du ] (8.8) u 8.5

6 Narrowband Noise Represenaion on Mac Equaion (8.8) becomes P e = erfc( A σ ) = erfc( A σ ) = erfc( A N ). Similarly, we can use his approach o derive he average probabiliy of error for BASK and BPSK sysems. In he BASK sysem, he synchronous deecor oupu consiss of a signal A plus noise or a noise alone. In he BPSK sysem, he synchronous deecor oupu consiss of a polar signal + A plus noise. The resuls are summarised in Table 8. BPSK P e = erfc( A N ) BFSK P e = erfc( A N ) BASK P e = erfc( A N ) References Table 8. Performance of various modulaion sysems. [] J. G. Porakis and M. Salehi, Communicaion Sysems Engineering, Prenice Hall, 994. ISBN [] M. S. Roden, Analog and Digial Communicaion Sysems, 3/e, Prenice Hall, 99, ISBN [3] H. Taub and D. L. Schilling, Principles of Communicaion Sysems, /e, MaGraw-Hill, 996, ISBN

7 Narrowband Noise Represenaion on Mac Wideband noise Bandpass filer f c Narrowband noise G n (f ) -f c f c Figure 8. Generaion of narrowband noise. x ( )+ x ( ) cos 4π f c - y ( ) sin 4 π f c LPF x ( ) n ( ) cos π f c LPF y ( ) -sin π f c Figure 8. Generaion of quadraure componens of n(). Gn ( f ) G x ( f ) = G y ( f ) G n ( f ) c G x ( ) = B -f c (a) B f c B B (b) Figure 8.3 (a) Power specral densiy of n(). (b) Power specral densiy of x() and y(). 8.7

8 Narrowband Noise Represenaion on Mac G n (f ) G x (f ) = G y (f ) G x () = G n (f c ) -f c (a) f c (b) Figure 8.4 (a) Power specral densiy of narrowband Gaussian noise n(). (b) Power specral densiy of x() and y(). r ( ) = Acos π f c + w ( ) r ( ) = BPF f c BPF f c A cos π f c + n ( ) LPF Gain = cos π f LPF Gain = cos π f v ( ) = + - A + x ( ) v = A + x v v = x v ( ) = x ( ) r ( ) = n ( ) Figure 8.5 Synchronous deecion of binary FSK signals. π σ (v + A) - e σ = Probabiliy densiy funcion f ( v /) f ( v /) = πσ (v - A ) - e σ -A A v Figure 8.6 Condiional probabiliy densiy funcion. 8.8