The performance of AM and FM receivers. Editor: Xuanfeng Li Teacher: Prof. Xiliang Luo
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1 The performance of AM and FM receivers Editor: Xuanfeng Li Teacher: Prof. Xiliang Luo
2 The performance of AM receivers using Envelop Detection In a full AM signal, both sidebands and the carrier wave are transmitted, as shown by ss AAAA tt = AA cc 1 + kk AAAA mm(tt) cccccccππff cc tt Assume mm tt = 0, ββ AAAA 1 After band-pass filter, the narrow band noise nn ii tt is modeled as white Gaussian noise of zero mean and power spectral density NN 0 /2 nn ii tt = nn cc tt cccccccππff cc nn ss tt ssssss2ππff cc tt
3 The performance of AM receivers using Envelop Detection Then we have power of signal and noise respectively SS ii = ss 2 mm (tt) = AA2 cc 2 mm 2 (tt), NN ii = nn 2 ii (tt) = NN 0 WW 2 + AA cc B is bandwidth of band-pass filter. The input SNR is SS ii = AA cc NN ii 2NN 0 WW The input of demodulator is ss AAAA tt + nn ii tt = AA cc 1 + mm(tt) + nn cc tt cccccccππff cc tt nn ss tt sssssssππff cc tt = EE tt cos[2ππff cc tt + φφ(tt)] where EE tt = AA cc 1 + mm(tt) + nn cc tt 2 + nn 2 ss (tt) nn ss tt φφ tt = arctan[ AA cc 1 + mm(tt) + nn cc tt ] In order to simplify the analysis, we consider two special case AA cc 2 kk AAAA 2 mm 2 (tt)
4 The performance of AM receivers using Envelop Detection (a)large SNR It means AA cc 1 + mm(tt) nn cc 2 tt + nn ss 2 (tt) Thus EE tt = AA cc mm(tt) 2 + 2AA cc [1 + mm tt ]nn cc tt + nn cc 2 (tt) + nn ss 2 (tt) AA cc mm tt 2 + 2AA cc 1 + mm tt nn cc tt AA cc 1 + mm tt 1 + 2nn cc tt AA cc 1+mm tt AA cc 1 + mm tt 1 + nn cc tt AA cc 1+mm tt = AA cc 1 + mm tt + nn cc tt Since (1 + xx) 1 + xx 2, xx 1
5 The performance of AM receivers using Envelop Detection After eliminating DC component, we have SS oo = ss mm 2 (tt) = AA cc 2 mm 2 (tt), NN oo = nn cc 2 (tt) = nn cc 2 (tt)=nn 0 WW The output SNR is SS oo = AA2 2 cckk AAAA mm 2 (tt) NN oo 2NN 0 WW Then we have GG AAAA = SS oo/nn oo SS ii /NN ii = AA2 2 cckk AAAA mm 2 (tt) AA 2 cc +AA 2 2 = cc kk AAAA mm 2 (tt) mm 2 (tt) 1+kk AAAA 2 mm 2 (tt) < 1 Since ββ AAAA 1. And if mm tt = AA mm cccccccππff mm tt, mm 2 tt = 1 2 AA mm 2 μμ2 GG AAAA =, µ = kk 2+μμ AAAAAA 2 mm When 100% modulation, which means µ = 1, we get 1/3. Thus we know that envelop detector lowers the SNR.
6 The performance of AM receivers using Envelop Detection (a)small SNR It means Then Where EE tt = AA cc 1 + mm(tt) nn cc 2 tt + nn ss 2 (tt) AA cc mm(tt) 2 + 2AA cc [1 + mm tt ]nn cc tt + nn cc 2 (tt) + nn ss 2 (tt) 2AA cc [1 + mm tt ]nn cc tt + nn cc 2 (tt) + nn ss 2 (tt) = [nn 2 cc (tt) + nn 2 ss (tt)]{1 + 2nn cc tt AA cc [1+mm tt ] } nn 2 cc (tt)+nn 2 ss (tt) = RR(tt) 1 + 2AA cc 1+mm tt RR tt ccccccθθ(tt) RR(tt) + AA cc [1 + mm tt ] ccccccθθ(tt) RR tt = nn 2 cc tt + nn 2 ss tt, θθ tt = arctan nn ss tt nn cc tt Now the signal becomes part of noise and the system will no longer work.
7 The performance of FM receivers The incoming FM signal is defined as, ss FFFF tt = AA cc cos[2ππff cc tt + 2ππkk FFFF mm ττ ddττ] Still assume mm tt = 0 Then we have power of signal and noise respectively SS ii = AA cc 2 2, NN ii = nn ii 2 (tt) = NN 0 WW B is bandwidth of band-pass filter. The input SNR is SS ii NN ii = AA 2 cc 2NN 0 WW
8 The performance of FM receivers (a)large SNR Ignore the interaction between signal and noise under large SNR. Assume nn ii tt = 0, mm oo tt = kk dd kk FFFF mm tt kk dd is discriminator index, let it be 1. Thus we have SS oo = mm oo 2 (tt) = (kk FFFF ) 2 mm 2 (tt) Assume m tt = 0, the input of demodulator is AA cc cccccccππff cc tt + nn ii tt = [AA cc +nn cc tt ]cccccccππff cc nn ss tt sssssssππff cc tt = AA tt cos[2ππff cc + φφ tt ] Envelop AA tt = [AA cc + nn cc tt ] 2 +nn 2 ss tt Since SNR is large, the phase nn ss (tt) φφ tt = aaaaaaaaaaaa AA cc + nn cc (tt) aaaaaaaaaaaa nn ss(tt) AA cc nn ss(tt) AA cc
9 The performance of FM receivers Thus the output noise is nn dd tt = ddφφ tt dddd = 1 ddnn ss (tt) 2ππAA cc dddd The power spectrum density is So we have PP dd ff = ( kk dd ) 2 HH(ff) PP AA ss ff = ( 1 ) 2 ff 2 NN cc AA 0, ff < BB FFFF cc 2 NN 0 = WW WW WW PP dd ff dddd = ( 1 ) 2 ff 2 NN 0 dddd = 2NN 0WW 3 AAcc WW 3AA cc 2
10 The performance of FM receivers SNR at the output SS OO = 3AA2 cckk 2 FFFF mm 2 (tt) NN OO 2NN 0 WW 3 Then GG FFFF = 3kk FFFF 2 mm 2 (tt) WW 2 Example : Single-tone Modulation The modulated FM signal is defined by ss FFFF (tt) = AA cc cos[2ππff cc tt + mm ff ssssss2ππff mm tt] Where mm ff = kk FFFF ωω mm = ωω ωω mm = ff ff mm So we get mm tt = ff kk FFFF cccccccππff mm tt
11 The performance of FM receivers And mm 2 tt We have SS OO = 3AA2 cc( ff) 2 NN OO 4NN 0 WW 3 Where ββ = ff/ww is modulation index = ( ff)2 2 2kk FFFF = 3AA2 ccββ 2 4NN 0 WW GG FFFF = 3ββ2 2 For an AM system operating with a single-tone signal and 100% modulation, GG AAAA = 1 3 We just need to adjust ββ 0.5, there will be GG FFFF >GG AAMM
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