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# Tele3113 wk6tue

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### Tele3113 wk6tue

1. 1. TELE3113 Analogue and Digital Communications Transmission of FM Waves Wei Zhang w.zhang@unsw.edu.auSchool of Electrical Engineering and Telecommunications The University of New South Wales
2. 2. Wideband FM SignalIn the last lecture, we studied: The FM wave of the single-tone message signal is given by ∞ s(t) = Ac Jn (β) cos[2π(fc + nfm )t]. n=−∞ The spectrum of s(t) is given by ∞ Ac S(f ) = Jn (β)[δ(f − fc − nfm ) + δ(f + fc + nfm )]. 2 n=−∞ TELE3113 - Transmission of FM Waves. August 25, 2009. – p.1/
3. 3. Transmission BW of FM (1) In theory, an FM wave contains an inﬁnite number of side-frequencies, so the BW is inﬁnite. In practice, the FM wave is effectively limited to a ﬁnite number of signiﬁcant side-frequencies.Speciﬁcally, by observing the spectrum, for large β, the BW approaches 2∆f = 2βfm . for small β, the BW approaches 2fm .Carson’s rule: 1 BT ≈ 2∆f + 2fm = 2∆f 1+ . β TELE3113 - Transmission of FM Waves. August 25, 2009. – p.2/
4. 4. Transmission BW of FM (2) Carson’s rule is simple, but not an accurate estimate. An accurate estimate of the BW can be deﬁned as the separation between the two frequencies beyond which none of the side frequencies is greater than 1% of the unmodulated carrier amplitude. Mathematically, the BW is given by (universal rule) nmax BT = 2nmax fm = 2∆f β where nmax is the largest value of the integer n that satisﬁes |Jn (β)| > 0.01. TELE3113 - Transmission of FM Waves. August 25, 2009. – p.3/
5. 5. Transmission BW of FM (3) β 0.1 0.3 0.5 1 2 5 10 20 302nmax 2 4 4 6 8 16 28 50 70 2 10BT 2nmax∆f = β BT/∆ f in term of β Normalized bandwidth, B /∆ f T 1 10 0 10 −1 0 1 10 10 10 Modulation index, β TELE3113 - Transmission of FM Waves. August 25, 2009. – p.4/
6. 6. Transmission BW of FM (4)Relative error of BW estimate with Carson’s rule compared tothe universal rule is deﬁned by Carson BT 1+β = 1 − Universal = 1− × 100%. BT nmaxUsually, Carson’s rule underestimates the BW by 25%. 40 35 Relative error of BW estimate (%) 30 25 20 15 10 Relative error of Carson’s rule in term of β 5 0 −5 TELE3113 - Transmission of FM Waves. August 25, 2009. – p.5/ −10 −1 0 1
7. 7. Generation of FM Waves (1)Armstrong’s method: The message signal is ﬁrst used to produce a narrow-band FM. Then, a frequency multiplication is used to produce a wide-band FM. Message Narrow- signal band phase Frequency Integrator multiplier m(t) Modulator Wide- band FM wave Narrow-band phase modulator Crystal controlled oscillator TELE3113 - Transmission of FM Waves. August 25, 2009. – p.6/
8. 8. Generation of FM Waves (2) Frequency multiplier FM wave FM wave s(t) s’(t) with with carrier Bandpass Memoryless v(t) filter with carrier frequency fc nonlinear frequency nfc and mid-band device frequency nfc and   modulation modulation   nThe input-output relation of the nonlinear device is v(t) = a1 s(t) + a2 s2 (t) + · · · + an sn (t),where a1 , a2 , · · · , an are coefﬁcients determined by the deviceand n is the highest order of nonlinearity. TELE3113 - Transmission of FM Waves. August 25, 2009. – p.7/
9. 9. Generation of FM Waves (3) Consider the input s(t) is an FM wave deﬁned by t s(t) = Ac cos 2πfc t + 2πkf m(τ )dτ . 0 Suppose that bandpass ﬁlter is designed to have a BW equal to n times the BW of s(t). After bandpass ﬁltering of the output v(t), we have t s (t) = Ac cos 2πfc t + 2πkf m(τ )dτ , 0 where fc = nfc and kf = nkf . TELE3113 - Transmission of FM Waves. August 25, 2009. – p.8/
10. 10. Demodulation of FM Waves Recall that the FM signal is given by t s(t) = Ac cos 2πfc t + 2πkf m(τ )dτ . 0 After taking the derivative of s(t) with respect to t, we get t ds(t) = −2πAc [fc + kf m(t)] sin 2πfc t + 2πkf m(τ )dτ . dt 0 The derivative is indeed a band-pass signal with amplitude modulation and the amplitude is 2πAc fc [1 + kf m(t)/fc ]. If fc is large enough such that the carrier is not overmodulated, then we can recover the message signal m(t) with an envelop detection. TELE3113 - Transmission of FM Waves. August 25, 2009. – p.9/