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Tele3113 wk4tue

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Tele3113 wk4tue

1. 1. TELE3113 Analogue and Digital Communications SSB Modulation Wei Zhang w.zhang@unsw.edu.auSchool of Electrical Engineering and Telecommunications The University of New South Wales
2. 2. Last week ...We have studied: Amplitude Modulation: s(t) = [1 + ka m(t)]c(t). Simple envelope detection, but low power efﬁciency. DSB-SC Modulation: s(t) = m(t)c(t). High power efﬁciency, but requiring a perfect phase recovery for coherent detection.Both AM and DSB-SC have two symmetric sidebands in themodulated wave, thereby causing the wastage of bandwidth. TELE3113 - SSB Modulation. August 11, 2009. – p.1/1
3. 3. From DSB-SC to SSB (1)First, let us review DSB-SC modulation of a single-tonemessage signal m(t) = Am cos(2πfm t). The modulated signal issDSB (t) = m(t)c(t) = Am Ac cos(2πfm t) cos(2πfc t) 1 1 = Am Ac cos[2π(fc + fm )t] + Am Ac cos[2π(fc − fm )t]. 2 2The FT of the DSB-SC modulated signal is given by Am Ac Am Ac SDSB (f ) = δ(f − fc − fm ) + δ(f + fc + fm ) 4 4 Am Ac Am Ac + δ(f − fc + fm ) + δ(f + fc − fm ). 4 4 TELE3113 - SSB Modulation. August 11, 2009. – p.2/1
4. 4. From DSB-SC to SSB (2)Suppose that we want to generate a sinusoidal SSB modulatedwave that retains the upper side-frequency at fc + fm . Bysuppressing the second term in the equation of s DSB (t), we get 1 sUSSB (t) = Am Ac cos[2π(fc + fm )t]. 2It can be further expressed as (using the trigonometric identitycos(x + y) = cos x cos y − sin x sin y) 1 sUSSB (t) = Am Ac cos(2πfc t) cos(2πfm t) 2 1 − Am Ac sin(2πfc t) sin(2πfm t). 2 TELE3113 - SSB Modulation. August 11, 2009. – p.3/1
5. 5. From DSB-SC to SSB (3)Suppose that we want to generate a sinusoidal SSB modulatedwave that retains the lower side-frequency at fc − fm . Bysuppressing the ﬁrst term in the equation of sDSB (t), we get 1 sLSSB (t) = Am Ac cos[2π(fc − fm )t]. 2We further express it as 1 sLSSB (t) = Am Ac cos(2πfc t) cos(2πfm t) 2 1 + Am Ac sin(2πfc t) sin(2πfm t). 2 TELE3113 - SSB Modulation. August 11, 2009. – p.4/1
6. 6. From DSB-SC to SSB (4)Combining the equations of sUSSB (t) and sLSSB (t), we get theSSB modulated wave of a single-tone message signalm(t) = Am cos(2πfm t) as follows: 1 sSSB (t) = Am Ac cos(2πfc t) cos(2πfm t) 2 1 Am Ac sin(2πfc t) sin(2πfm t), 2where the minus and plus signs apply to the upper SSB andlower SSB, respectively. TELE3113 - SSB Modulation. August 11, 2009. – p.5/1
7. 7. SSB For a periodic message signal m(t) = m am cos(2πfm t), the SSB modulated wave is 1 sSSB (t) = Ac cos(2πfc t) am cos(2πfm t) 2 m 1 Ac sin(2πfc t) am sin(2πfm t). 2 m Generally, for a Fourier transformable message signal m(t), the SSB modulated wave is 1 1 sSSB (t) = Ac m(t) cos(2πfc t) Ac m(t) sin(2πfc t), ˆ 2 2 ˆ where m(t) is Hilbert transform of m(t). (See next page) TELE3113 - SSB Modulation. August 11, 2009. – p.6/1
8. 8. Hilbert Transform (1) ˆ The signal m(t) is the Hilbert transform of the signal m(t), deﬁned as 1 ∞ m(τ ) m(t) = ˆ dτ π −∞ t − τ 1 = m(t) . (convolution) πt If m(t) ⇔ M (f ), then ˆ m(t) ⇔ M (f ) = −jsgn(f )M (f ), ˆ   1, f > 0    where the sign function is sgn(f ) = 0, f = 0    −1, f < 0  TELE3113 - SSB Modulation. August 11, 2009. – p.7/1
9. 9. Hilbert Transform (2) 1 m (t ) h (t ) = m(t ) ˆ πt Illustration of Hilbert transform in time domain M(f ) H ( f ) = − j sgn( f ) ˆ M(f ) Illustration of Hilbert transform in frequency domain TELE3113 - SSB Modulation. August 11, 2009. – p.8/1
10. 10. Hilbert Transform (3)Note that the frequency response of Hilbert transformer 1h(t) = πt is H(f ) = −jsgn(f ).The magnitude of H(f ) is given by   1, f > 0 |H(f )| =  1, f < 0and the phase is given by   −90◦ , f > 0 ∠H(f ) =  90◦ , f < 0 TELE3113 - SSB Modulation. August 11, 2009. – p.9/1
11. 11. Spectra of SSBFor positive frequencies, the spectra of the two kinds of SSBmodulated waves are as follows: For the upper SSB,   Ac M (f − f ), f ≥ fc 2 c S(f ) =  0, 0 < f < fc For the lower SSB,   0, f ≥ fc S(f ) =  Ac M (f − fc ), 0 < f < fc 2 TELE3113 - SSB Modulation. August 11, 2009. – p.10/1
12. 12. Modulation of SSB (1)Frequency Discrimination Method Message signal SSB-Modulated m (t ) Product Band-pass signal s (t ) modulator filter Ac cos( 2πf c t ) Carrier wave TELE3113 - SSB Modulation. August 11, 2009. – p.11/1
13. 13. Modulation of SSB (2)Phase Discrimination Method + Message signal SSB-Modulated m(t ) Product signal s (t ) ∑ modulator cos(2πf c t ) m Oscillator Wideband − 900 Phase-shifter Phase-shifter sin(2πf c t ) ˆ m (t ) Product modulator TELE3113 - SSB Modulation. August 11, 2009. – p.12/1
14. 14. Demodulation of SSB Modulated Demodulated wave s (t ) v(t ) signal v o (t ) Product Low-pass modulator filter Ac cos(2πf c t + φ ) Local oscillator Suppose in the receiver the local oscillator can provide the same frequency, but arbitrary phase difference φ, measured with respect to the carrier wave c(t). It applies equally well to the demodulation of both DSB-SC and SSB. TELE3113 - SSB Modulation. August 11, 2009. – p.13/1