Intermediate Frequency (IF)
Image frequency
ERG2310A-II p. II-33 Amplitude Modulation: SSB
DSB modulation:
ω By mixing with a sinusoidal carrier at c rad/sec, half of this spectral ω density is translated up in frequency and centered about c and half is ω translated down to (- c ). Î Doubling of the bandwidth of a given signal
Since each half contains all the information about the signal, the original signal can be recovered again from either the upper or lower pair of sidebands by an appropriate frequency translation. So only the upper or the lower pair of sidebands is required to transmit.
Such kind of modulation is called single-sideband (SSB) modulation. It is efficient because it requires no more bandwidth than that of the original signal and only half of the corresponding DSB signal.
ERG2310A-II p. II-34 Amplitude Modulation: SSB
SSB:
SSB:
ERG2310A-II p. II-35 Amplitude Modulation: SSB
Generation of SSB Signals -first generate a DSB signal, then suppress one of the sidebands by filtering.
ERG2310A-II p. II-36 Amplitude Modulation: SSB
ω Generation of SSB: Phase-Shift Method ℑ{e j mt } jω t Considerω the modulating signal x(t) is x(t) = e m ω ω ω j ct 0 ω ω and let the carrier signal be m e . ω ω ω ℑ j ct ω {e } j ct = j mt j ct Multiplying, we get x(t)e ω e e ω ω ω 0 c Taking the real part,ω we have ω ω ℑ{e j ct e j mt } ω ω Re{e j mt e j ct } = Re{e j mt }Re{e j ct } − Im{e j mt }Im{e j ct } ω ω ω ω ω = − ω ω 0 c c+ m cos mt cos ct sin mt sin ct ω ℑ{Re[e j ct e j mt ]} Because this represents the upper sideband, we write = ω ω − ω ω sSSB+ (t) cos mt cos ct sin mt sin ct ω −ω -ω 0 ω ω +ω − ω c m c c m Similarly, by using x ( t ) = e j m t , the lower sideband is = ω ω + ω ω sSSB− (t) cos mt cos ct sin mt sin ct
In general, we can write s (t) = x(t)cosω t ± xˆ(t)sinω t SSBm c c
where xˆ(t) is that signal obtained by shifting the phase of x(t) by 90° at each frequency. ERG2310A-II p. II-37 Amplitude Modulation: SSB
Generation of SSB Signals : Phase-Shift Method
ERG2310A-II p. II-38 Amplitude Modulation: SSB
Demodulation of SSB Signals The synchronous detector will properly demodulate SSB-SC signals
ERG2310A-II p. II-39 Amplitude Modulation: SSB
s (t) = x(t)cosω t ± xˆ(t)sinω t Given that the incoming SSB-SC signal is SSBm c c = ω + ∆ω +θ Let the locally generated carrier signal be c(t) cos[( c )t ], ω where (∆ω) is the frequency errorθand θ is the phase error. ω ω s (t)c(t) = [x(t)cosω t ± xˆ(t)sinθ ω t]cos[(ω + ∆ω)t +θ ] SSBm c c c ω = 1 x(t){cos[(∆ )t + ] + cos[(2ω + ∆ )t +θ ]} 2 c ω 1 ∆ + − + ∆ +θ m 2 xˆ(t){sin[( )t ] sin[(2 c )t ]}
After passing through a low-pass filter, the output xo(t) becomes
= 1 ∆ω +θ 1 ∆ω +θ Distorted ! xo (t) 2 x(t)cos[( )t ] m 2 xˆ(t)sin[( )t ]
= 1 If ∆ω=0 and θ =0, then xo (t) 2 x(t)
ERG2310A-II p. II-40 Amplitude Modulation: SSB
Example: When an SSB (upper/lower-sideband) is received and fed into the (SSB-SCfollowing at 20 MHz) demodulator:
sSSB±(t)IF Amp LPF xo(t) (10.000-10.003 MHz) π π cos(2 fct) cos(2 fdt) -f -f +f +f c d d 3kHz c fc=30.003MHz Upper sideband f =10.003MHz SSB-SC d -20 MHz 0 20 MHz 10.003 MHz 10 MHz
-fc -fd +fd +fc Freq. fc=30.000MHz Lower sideband f =10.000MHz SSB-SC 3kHz d -20 MHz 20 MHz 0 10.003 MHz 10 MHz
ERG2310A-II p. II-41 Freq. Amplitude Modulation: SSB
Single Sideband-Large Carrier (SSB-LC) Signals An expression for an SSB-LC signal is = ω + ω ω s(t) Ac cos ct x(t)cos ct m xˆ(t)sin ct The original signal x(t) can always be recovered from s(t) using synchronous detection. If the carrier is large, however, envelope detection can also be used. The envelope can be written as 2x(t) x 2 (t) xˆ 2 (t) = + 2 + ˆ 2 = + + + r(t) [Ac x(t)] [x(t)] Ac 1 2 2 Ac Ac Ac If the carrier is much larger than the SSB-SC envelope, we have ≈ + 2x(t) ≈ + x(t) = + r(t) Ac 1 Ac 1 Ac x(t). Ac Ac Thus after discarding the dc term introduced by carrier, the SSB-LC signal can then be demodulated correctly using an envelope detector.
ERG2310A-II p. II-42 Amplitude Modulation: VSB
Vestigial sideband (VSB) modulation is a compromise between DSB and SSB. In VSB modulation, one passband is passed almost completely whereas only a residual portion of the other sideband is retained in such a way that the demodulation process can still reproduce the original signal. ω SDSB( )
−ω 0ωω c ω c SVSB( )
−ω 0ωω c c The partial suppression of one sideband reduces the required bandwidth from that required for DSB but does not match the spectrum efficiency of SSB.
If a large carrier is also transmitted, the desired signal can be recovered using an envelope detector. If no carrier is sent, the signal can be recovered using a synchronous detector or the injected carrier method. ERG2310A-II p. II-43 Amplitude Modulation: VSB
Generation of VSB Signals
ω = 1 ω −ω + 1 ω +ω ω SVSB( ) [2 X( c ) 2 X( c )]H( )
The filtering operation can be represented by a filter H (f) that passes some of the lower (or upper) sideband and most of the upper (or lower) sideband. ERG2310A-II p. II-44 Amplitude Modulation: VSB
Demodulation of VSB Signals
sVSB(t) LPF xo(t)
ω Synchronous Demodulation cos ct The spectral density of the received vestigial-sideband signal is
ω = 1 ω −ω + 1 ω + ω ω SVSB ( )ω [ 2 X ( c ) 2 X ( c )]H ( ) ω The output of the synchronousω detector is ω x (t) = [s ω(t)cosω t] ω o VSB ω c LP ω= 1 []+ ω + 1 ω []− X o ( ) 4 X ( ) H ( c ) LP 4 X ( ) H ( c ) LP ω ω ω = 1 []+ + −ω ω 4 X ( ) Hω( c ) H ( c ) LP For faithful reproduction of x(t), we require that ω − + + = ≤ ω [H ( c ) H ( c )]LP constant, m .
ERG2310A-II p. II-45 ω ω ω ω Amplitude Modulation: VSB ω − + + = ≤ ω [H ( c ) H ( c )]LP constant, m .
ω ω ω ω ω By letting the constant be ω : 2H( c) ω − + + = ≤ ω [H ( c ) H ( c )]LP 2H ( c ), m ω ω Thus, H( ) exhibits odd symmetry around the carrier frequency c. The sum of the values of H(ω) at any two frequencies equally ω displaced above and below c is unity.
ERG2310A-II p. II-46 Amplitude Modulation: VSB
Synchronous Demodulation of VSB Signals
Received ω VSB signal SVSB( )
−ω 0 ω c c ω After mixer Xd( ) ω
−2ω −ω 0 ω 2ω c c c c
ω After LPF Xo( ) ω
−ω 0 ω c c
ω
ERG2310A-II p. II-47 Amplitude Modulation: VSB
Example of VSB Signals: Television picture signal has nominal bandwidth of 4.5MHz If DSB modulation is used,Television it requires signal at least 9MHz for each TV channel. So, VSB modulation is used so that the whole TV signal is confined to about 6MHz.
ERG2310A-II p. II-48 Angle Modulation
A continuous-wave (CW) sinusoidal signal can be varied by changing its amplitude and its phase angle. = ω + φ s(t) A(t)cos[ ct (t)] To carry a message signal x(t): Amplitude modulation: Keep θ(t) constant and varies A(t) proportionally to x(t). = []ω + φ ∝ s(t) A(t)cos ct o where A(t) x(t) Angle modulation: ω φ Keep A(t) constant and varies [ ct+ (t)] proportionally to x(t).
= [][]ω + φ ω + φ ∝ s(t) Acos ct (t) where ct (t) x(t)
ERG2310A-II p. II-49 Angle Modulation
Phasor Representation The phasor representation of a constant-amplitude sinusoid is shown as follows
ω A: magnitude of the phasor i(t) θ(t): phase angle θ (t) = ω t + φ(t) +θ θ(t) c o ω i(t) : instantaneous angular rate dθ (t) dφ(t) ω (t) = = ω + θi dt c dt ω τ = t τ +θ (t) i ( )d 0 ∫0
ERG2310A-II p. II-50 Angle Modulation
If the phase φ(t) is varied linearly with the input signal x(t) , we have θ = ω + +θ φ ∝ (t) ct k p x(t) 0 i.e. (t) x(t)
ω θ where c , kp , 0 are constants.
As the phase is linearly related to x(t), this type of angle modulation is called phase modulation (PM) with
= ω + +θ sPM (t) Ac cos[ ct k p x(t) o ]
and , φ(t) is called instantaneous phase deviation. ω dθ (t) dφ(t) dx(t) ω (t) = = ω + ⇒ = ω + k i dt c dt i(PM) c p dt
ERG2310A-II p. II-51 Angle Modulation
ω Ifω the instantaneous frequency i proportional to the input signal, we have φ θ = ω + d (t) ∝ dθ (t) dφ(t) i(FM ) (t) c k f x(t) ⇒ x(t) ω (t) = = ω + ω dt Q i dt c dt
t τ τ = t + k x( )d +θ (t) c f ∫ o 0 ω where c , kf are constants. As the frequency is linearly related to x(t), this type of angular modulation is called frequency modulation (FM)ω , with
t τ τ s (t) = A cos[ t + k x( )d +θ ] FM c c f ∫ o 0 dφ(t) and; is called instantaneous frequency deviation. dt
ERG2310A-II p. II-52 Angle Modulation
Phase Modulation (PM) Frequency Modulation (FM)
ω Instantaneous =ω + dx(t) ω (t) = ω + k x(t) i(PM) c k p i(FM ) c f angular rate dt ω
Modulated t τ τ signal = ω + +θ s (t) = A cos[ t + k x( )d +θ ] sPM (t) Ac cos[ ct k p x(t) o ] FM c c f ∫ o 0
Proportionality φ d (t) ∝ φ(t) ∝ x(t) x(t) dt
ERG2310A-II p. II-53 Angle Modulation
FM and PM Waveforms
ERG2310A-II p. II-54 Angle Modulation: Fourier spectra
Consider an angle-modulated signal = []ω + φ s(t) Ac cos ω ct (t) φ []+ = Re{}A e j ct (t) c ω φ {}φ φ = j ct j (t) Re Ac e e
jφ(t) ω φ Expand e in a powerφ series, gives ω 2 n ω (φt) (t) = j ct + − − + n + s(t) ReAc e 1 j (t) L j L 2! ω nφ! 2 3 = − − (t) + (t) ω + Ac cos ct (t)sin ct cos ct sin ct L 2! 3!
The signal consists of an unmodulated carrier plus various amplitude- φ ω φ2 ω φ3 ω modulated terms, such as (t)sin ct, (t)cos ct, (t)sin ct, …, etc. Hence the Fourier spectrum consists of an unmodulated carrier plus φ φ2 φ3 ω spectra (sidebands) of (t), (t), (t), …, etc., centered at c.
ERG2310A-II p. II-55 Angle Modulation: Narrowband
φ If | (t)|max << 1 , then, by neglecting the higher-power terms of φ(t) in the s(t) , gives ≈ ω − φ ω s(t) Ac cos ct Ac (t)sin ct
which is called the narrowband (NB) angle-modulated signal. ω ≈ ω − ω sNBPM (t) Ac cos ct Ac k p x(t)sin ct τ t τ s (t) ≈ A cos t − A k x( )d sinω t NBFM c c c f ∫ c 0
ERG2310A-II p. II-56 Angle Modulation: NBFM (Sinusoid)
Narrowband FM (NBFM) = ω Consider x(t) am cos mt ω = ω + For FM, i c k f x(t) = + ω c am k f cos mt
where kf is the frequency modulation constant; typical units are in radians per second per volt.
Define a new constant called the peak (maximum) frequency deviation, ∆ω = am k f thus, we have ω = ω + ∆ω ω i c cos mt
ERG2310A-II p. II-57 θAngle Modulation: NBFM (Sinusoid) ω The phaseτ of this FM signal is τ θ = t + (t) i ( )d 0 ω∫0 ω = t + ∆ ω + [ c cosτ m ]d 0 ω ∫0 ω τ ∆ θ = t +ω sinω t + 0 θ is set to zero for convenience. ω c m 0 β mω ∆ω = t + sinω tβ where β = c ω m ω ω m Thus, the resulting FM signalβ is ω = { jθ (t) } sFM (t) Re Ac e = A cos( t + sin t) ω c c m β = − ω Ac cos ct cos( sin mt) Ac sin ct sin( sin mt) For narrowband FM (NBFM) , β is very small so that β ω ≈ β ω ≈ β ω cos( sin mt) 1, sin( sin mt) sin mt Thus, = ω − β ω ω sNBFM (t) Ac cos ct Ac sin mt sin ct The parameter β is called the modulation index of the FM signal. ERG2310A-II p. II-58 Angle Modulation: NBPM (Sinusoid)
Narrowband PM (NBPM) = ω Consider x(t) am sin mt The phase of this PM signal is θ =ωω + +θ (t) ct k p x(t) 0 = + + θ ω ct k p am sinω mt 0 0 is set to zero for convenience. β ω = + ω ct sin mt β = k a ω ω p m The resulting PM signal is β = { jθ (t) } ω sPM (t) Re Ae = A cos( t + k asin t) ω c c p m β = − ω Ac cos ct cos( sin mt) Ac sin ct sin( sin mt) For narrowband PM (NBPM), β is very small, β ω ≈ β ω ≈ β ω cos( sin mt) 1, sin( sin mt) sin mt Thus, = ω − β ω ω s NBPM (t) Ac cos ct Ac sin mt sin ct ERG2310A-II p. II-59 Angle Modulation: NB (Sinusoid)
In summary, if the message signal x(t) is a pure sinusoid, that is, a sinω t for PM x(t) = m m ω am cos mt for FM
Then, φ = β ω (t) sin mt β k p am for PM = k a ∆ω where ωf m = ω for FM m m ∆ω : peak frequency deviation
Note that β is known as modulation index for angle modulation and is the maximum value of phase deviation for both PM and FM. It is only defined for sinusoidal modulation.
φ If (t) has a bandwidth of WB, the NB angle-modulated signal will have a bandwidth of 2WB.
ERG2310A-II p. II-60