Amplitude-Modulation System

A Method for frequency Translation

Frequency translation is a process which translates the original spectral range of a signal to a new spectral range by multiplying the signal with a sinusoidal signal. If the signal is band-limited to the frequency range extending from a frequency f1 to f2 then after frequency translation the new signal has a spectral range extends from f’1 to f’2 and the new signal bears the same information as was carried out by the original signal.

Example:Consider a signal Vm(t)=Am coswmt=Amcos 2**fm*t

=> )(2

)( tjwtjwmm

mm eeA

tV

Where Am is the constant amplitude and fm=wm/2 is the frequency of Vm(t).The two sided spectrum of Vm(t) is as shown in Fig.1

Am/2

-fm 0 fm f

Fig. 1: Two sided amplitude spectrum of Vm(t)=Am coswmt

Let consider another signal Vc(t)=Ac coswct= Accos 2**fc*t = )(2

)( tjwtjwcc

cc eeA

tV

Where Ac is the constant amplitude and fc=wc/2 is the frequency of Vc(t).The two sided spectrum of Vc(t) is shown in fig 2.

Ac/2

-fc 0 fc f

Fig. 2: Two sided amplitude spectrum of Vc(t)=Ac coswct

Multiplying the above two signals we will get a new signal i.e

)cos()(4

)()()( mcmccm

cm wwwwCosAA

tVtVtV

= twwjtwwjtwwjtwwjcm mcmcmcmc eeeeAA )()()()(

4

The new signal is a translated signal of Vm(t) whose amplitude spectrum is shown in fig3.

AmAc/4

-fc-fm -fc -fc+fm 0 fc-fm fc fc+fm f

Fig 3: Amplitude spectrum of the product signal V(t)

From Fig. 3 we could observe that the two original spectral lines in fig. 1 have been translated, both in the positive frequency direction by amount fc and also in the negative frequency direction by the same amount fc.

Example 2:Consider an arbitrary signal m(t) whose two sided amplitude spectrum is as shown in fig 4.

Frequencyfm-fm

Fig 4: The amplitude spectrum of m(t)(the base band signal)

Multiplying m(t) with a sinusoidal signal )(2

)( tjwtjwcc

cc eeA

tV we will get translated signal

known as V(t)=m(t)*Vc(t) whose specrum will be like as shown in fig. 5

Fig. 5: The amplitude spectrum of V(t)=m(t)*Vc(t) (the band pass signal)

Frequency

BasebandBandwidth

fc

Passband Bandwidth

BasebandSpectrum

PassbandSpectrum

Upper SidebandLower Sideband

fm-fm fc+fmfc-fm

Fig. 6: Baseband becomes Passband by translation to higher frequencyThe positive frequency spectrum becomes the upper side-band and the negative frequency spectrum become the lower side band

Baseband Signal – The magnitude of the spectral components of a baseband signal are nonzero for frequencies near the origin (i.e., f=0) and negligible elsewhere. The information signal is called the baseband signal. The bandwidth is always a positive quantity so the bandwidth of this signal is fm. It is also known as complex envelop of the passband signal.

Passband Signal – The spectral magnitude of a bandpass signal has nonzero for frequencies in some band concentrated about a frequency f=±fc (where fc>>0 and known as carrier) and negligible elsewhere.

The multiplication of the baseband signal with a sinusoid carrier signal translates the baseband frequency band up to fc. This signal is known as passband signal. This passband signal’s spectrum extends in range from (-fc - fm ) to (fc + fm.). Hence, the passband signal bandwidth is double that of the baseband signal.

Need for Frequency Translation

The following points are few useful purposes of frequency translation,

-fc -fc -fc+ Frequency fc

Passband Bandwidth

fm-fm fc+fmfc-fm

Practicability size of antennas:When the free space is used as channel antennas are used for transmitting and receiving signals. For efficient transmission and reception the dimension of antennas should be proportional to the wavelength of the signal to be transmitted or received.

L , and f

1 hence if the frequency is low then the size of antenna should

be very high for efficient transmission and reception.

For Example consider a signal having frequency of 1KHz then the =300,000m hence the required antenna length should approximately 300KM which is totally impractical. Where as if we will multiply the 1Khz signal with a sinusoidal signal having frequency of 300Mhz then the frequency translated signal has a highest frequency of 300.001 Mhz and the minimum length of the antenna should be approximately 3m which is practically possible to design.

Frequency multiplexing:Frequency translation helps in transmitting many signals in a signal channel which is know as frequency division multiplexing (FDM) . Most of the time at the transmitter there are more than one signals may have same spectral range need to be transmitted through the same channel. In order to achieve this each signal is frequency translated to different non overlapping frequency ranges. At the receiver signals are recovered by band-pass-filters followed by recovery circuit.

Narrow-banding:Frequency translation helps in decreasing the ration of highest to lowest frequency component in a signal. If the ratio is small (narrow) then this processes is know a narrow banding. Narrow-banding makes the antenna suitable for both the end of the frequency range.

Let consider an speech signal extends from 300Hz to3.4 KHz then the ratio of highest to lowest frequency is 3.4KHz/300Hz=11.333. Here the antenna suitable for one end may be too long or too short for the other end. If the same speech signal is translated by a sinusoidal signal of 10MHz the new signal has a frequency range of 10.0003MHz to 10.0034 MHz. Then the highest to lowest frequency ratio is 1.00031. The antenna suitable for use at one end of the range would be same as the other end of the rage.

Common processing:Frequency translation helps to use a single processing apparatus for different frequency range signals having similar characteristic. Example: Radio receiver.

Amplitude ModulationThe baseband/message/information signals are transmitted to a long distance through the different channel using different modulation techniques. There are basically two kinds of modulations i.e. i) Amplitude, and ii) Angle modulation. The angle modulation is again categorized as i)Frequency, and ii)phase modulation. In this section we will concentrate on amplitude modulations and its variants.

Amplitude Modulation: In amplitude modulation technique, the amplitude of a high frequency sinusoidal signal (Carrier signal) is varied according to the instantaneous amplitude of the based band or message or information signal.

The amplitude modulator has two inputs i.e message siganal m(t) and the carrier signal c(t) (local oscillator high frequency sinusoidal signal) and one output that is the amplitude modulated signal. The simple AM modulator is a product modulator as shown in Fig. 1 produce the product signal as well as the individual input signals (assuming a constant gain). The spectrums of input and output signals are shown in Fig. 2 for illustration.

Fig. 1 : Block diagram of a product modulator.

Fig. 2 Spectrum of the m(t)

Fig. 3: Amplitude Spectrum of S(t)= |S(jw)|

From Fig. 3 it is clear that the multiplier output has the translated signal plus the carrier signal plus the original modulating signal. The original signal is a very low frequency signal which need not required to transmitting through the antenna. Hence a Band pass filter is

f-

M(jw)

f-

|S(jw)|

fm-fmfc+fmfc-fm fc-fc-fm -fc -fc+fm

k1(cmax/2)

k1(mmax/2)

c(t)

m(t)S(t)=k1m(t)+ k1c(t)+ k2m(t)c(t)

required at the end of the multiplier to block the based band signal. Hence the AM block diagram is a multiplier followed by a bandpass filter as shown in Fig. 4 and the corresponding amplitude spectrum of the AM signal is shown in Fig. 5.

Fig. 4: Block diagram of the amplitude modulator

Fig. 5: The spectrum of an AM signal.

The spectrum of AM signal in Fig. 5 is the translated baseband signal spectrum to the frequency fc and the carrier spectrum at fc. The bandwidth of AM signal if 2fm =((fc+fm)-(fc-fm)), which is the twice the bandwidth of the based band signal fm.

Mathematical representation of AM signal:

Case 1: If the base band signal is a sinusoidal signal i.e m(t)=AmCos(wmt)

Let the carrier signal is c(t)=AcCos(wct)Then the AM signal is S(t)=k1Ac Cos(wct)+ k2AmCos(wmt) Ac Cos(wct)

=Ac[k1+k2(Am/Ac)Coswmt]Coswctk2 is known as sensitivity of the modulator.

= Ac[1+mCoswmt]Coswct; if k1=1 and m=k2(Am/Ac)If k2=1 or not mentioned then m=Am/Ac.

Hence, S(t)= Ac Cos wct+ Ac m/2(cos(wc+wm)t+cos(wc-wm)t)Then, the FT of S(t)= S(jw) and,

c(t)

m(t)

S(t)= c(t)+ k m(t)c(t)=[1+k m(t)]c(t))BPF

f-

|S(jw)|

fc+fmfc-fm fc-fc-fm -fc -fc+fm

k1(cmax/2)

k1(mmax/2)

The m is an integer and known as modulation index. The modulation index is represents the depth of modulation. The value of modulation index affects the modulated signal shape and characteristics.

If the modulation index is greater than 1(m>1) then over modulation occours.

Case:2 If the base band signal is an arbitrary signal m(t)Let the carrier signal is c(t)=AcCos(wct)Then the AM signal is vam(t)=Ac[1+ m(t)]Coswct ; |m(t)| is the modulation index.

The FT of vam(t) is

Fig:6: Time domain low frequency information signal (arbitrary baseband signal m(t)

Fig.7 Output signal of a product modulator, the envelope of which is the information signal

1 2 3 Frequency0 4

BandwidthFig.8 The single side Baseband Spectrum.

Frequencyfm-fm

Fig. 9 Two sided Base Band Spectrum.

Frequency fc

Passband Bandwidth

fm-fm fc+fmfc-fm

Fig.10 : Spectrum of AM signal. DSB-WC (Double side band with carrier)

DSB-C Switching Modulator

Fig.11: An AM (DSB-C) Modulator

Consider the diode in the above circuit is an ideal diode and the carrier is stronger than message signal. The diode conducts when the combined signal (m(t)+Acos wct) is positive. Since, carrier is stronger than message signal the switch is regulated by the carrier only.

Hence the diode will on for half of the carrier time and off for half of the carrier period. The switching signal is a periodic signal as shown below with a fundamental period of TC.

Fig.12: The switching action of the diode

Hence, using Fourier series the S(t) can be represent as

The signal at the input to the BPF is y(t)=[m(t)+A cos wct]S(t).Hence,

+ -

BPF

m(t)

A Cos wct

y(t)

k1 coswct+k2m(t) cos wctR

S(t)

The 1st term of Y(t) is the base band , the second term is the carrier, the third term is the DSB-SC, the fourth term has a DC plus a carrier having 2fC, and the other terms are higher harmonics

of carrier. The Bandpass filter that passes wc±wm will pass which is

the AM or DSB-C signal.

DSB-C Demodulator

Envelope Detector

The envelope of a signal is its maximum value over a set sampling period. A diode circuit shown below also known as diode detector or envelope detector detects the envelope of AM signals is the simplest and cheapest method of demodulating AM signals. The diode detector detectors efficiency depends on the strength of the noise interference the AM signal. Excessive amount of noise causes severe envelope fluctuations and makes this method less effective. Additive noise distorts the amplitude of AM signal hence, AM radio’s vulnerability to noise and other atmospheric perturbations.

CR Vo

D

Vi

Fig. 13: – RC-Diode Circuit used for Envelope Detection

The envelope detector is consists of a diode, a resistor and a Capacitor as shown above. The AM signal is applied to the input terminals of the circuit. The Diode conducts as the voltage (amplitude) increases and the capacitor charges up. Now as the input voltage begins to go down compared to the charging voltage of the capacitor the Diode is reversed biased and capacitor discharges through the resistor. The cycle continues and each charge of the capacitor indicates the maximum value over that period. In fact the capacitor discharges slightly between cycles as shown in the figure below but this can be compensated for easily.

Fig. 14: – Response of Diode detector to the received AM or DSB-C signal.

The time constant RC is selected so that the change in capacitor voltage (VC) between cycles is at least equal to the decrease in input signal amplitude (AM signal) between cycles.

In fact for the efficient tracking of the envelope the following conditions should satisfied:1. RsCTc ( charging time constant should be less than or equal to the carrier frequency);

where Rs is the source resister. 2. TcRLC(1/fM) (Discharging time constant should be in between the carrier time period

and the 1/fM.); for following the maximum rate of change of the envelope.

Square-Law Demodulator

Handling non-linearity communication is a difficult task because it distorts the signal and produces unwanted products. Whereas non-linearity has one of its advantages is that it is used in demodulation of some AM signals.

Let’s take a non-linear device with the following behavior.

y kx 2

Fig. 15: Illustrating the operation of square-law demodulator. The output is the value of y averaged over many carrier cycle. (Adapted from PCS, H.Taub & D.L.Schilling)

Now let’s take an amplitude modulated signal

x t A m t tc c( ) [ ( )] cos 1

Putting this through the above non-lienearity, after some manipulations and clever trigonometric substitutions, we get

Now throw away the DC term, filter out the terms at two times the frequency and what we have left is

y t kA m tk

A m tc c( ) [ ( ) ( )] 2

2

The term m t2 ( ) is not a big problem if the modulation index is small. This term disappears and for audio broadcasting this term makes no discernible difference.The distortion may be small if i.e 2.

Hence, any type of nonlinearity which does not have odd-function symmetry with respect to the initial operating point can be used for demodulation of DSB-C signal.

Power of AM Signal (DSB-C):

Carrier Term + side band term

Hence the power of DSB-C signal is the sum of carrier power and sideband power

Carrier Power=

and the side band power=

Where, is the mean square value of m(t)

Therefore the total transmitted power of DSB-C is

PT=Pc+Ps=

y t x t kA kA m tk

A m t t kAk

A m tc c c c c c( )=k ( ) [ ( ) ( )] cos [ ( )] 2 2 2 2 2

22

2

DC term Information signal

Signal at 2wc

The sidebands contain the information where as the carrier does not contain any information. Thus the percentage of the total power carried by the side bands is known as the transmission efficiency ().

If m(t)=Am then =

Then

Where m is the modulation index kaAm.

will be maximum when m=1 (i.e 100% modulation).Then the max=[1/(2+1)]x100=33.33%.

Hence, in case of DSC-C for m=1, the efficiency is maximum i.e max=33.33%.. Thus, 67% of the total power is carried by the carrier which is a waste. For m<1 the efficiency is less than 33.33%.Example:

1. A carrier is modulated to a depth of 75%. Calculate the total power in the modulated wave. Assume the modulating signal to be sinusoid.

2. A broadcast radio transmitter radiates 5KW power when the modulation percentage is 60%. How much is the carrier power?

Current calculationPT=PC(1+(m2/2)=>IT

2= IC2(1+(m2/2)=> IT= IC [(1+(m2/2)]1/2

Single Side-Band AM signal (SSB-SC)In DSB-SC or DSB-C, there are two side bands and both having the same bandwidth. Both the side bands USB or LSB carry the same information. Hence it is possible to recover the information signal from the one side band i.e either USB or LSB. The advantage of transmitting a single side band is the bandwidth conservation because the bandwidth of SSB signal has half of the DSB-SC or DSB-C.

There are two methods of SSB Generations:1. Filtering Method2. Phasing Method.

Filtering Method:

The simplest solution would be to just take the DSB-SC signal and filter the unwanted band before transmission so that the unwanted side is not sent at all as shown in the figure below. By keeping only the part shown, we have gotten rid of all the other images, all of the negative components and the upper side-band.

Fig. 16: A passband filter after DSB-SC modulation results in getting rid all but one band.Problem with this method is that it is hard to build practical filters with steep enough cut-offs at high frequencies. Such a filter ends up distorting the desired signal as well as including some of the unwanted side-band anyway.For example: If the carrier frequency is 10 MHz then the selectivity of the band pass filter to generate a SSB signal in case of speech signal is (600/107)x100=0.006 percentage. (600 Hz is the allowed gap between the upper and lower side band in case of speech signal). Such a low selectivity filter is practically hard to build. Hence, the SSB generation is made by more than one stage of frequency translation and filtering to increase the selectivity of the BPFs as shown in Fig. 17.

Fig. 17: Block diagram of SSB generation using Filtering method. (Adapted from PCS, H.Taub & D.L.Schilling)

Frequency fc-fc+fm-fc-fm fc+fmfc-fm

Filter and keep

In the first BM used a carrier of 100KHz hence the selectivity of the filter is (600/100K)x100 =0.6 percentage. The USB band is 100.3KHz to 103KHz which is input to the second BM having the required carrier of 10MHz. Now the second BPF selectivity should be such that it should provide a 40 dB attenuation between 200.6KHz band at 10MHz frequency i.e (200.6K/10M)x100=2 percentage. This selective is very high in compared to one stage filtering method (0.006%).

Phasing Method:

Fig.18: SSB generation using Phasing Method.

In Phasing method there are two balance modulators and two 900phase shift networks used to generate the SSB signal. The inputs to the 1st BM are the base-band signal m(t) and the carrier signal cos(wct) where as the inputs to the 2nd BM are (900 phase shift of m(t)) and sinwct (900 phase shift of cos(wct)). Where is the Hilbert Tranform of m(t) and sinwct is the Hilbert transform of the cos(wct). ( is the pi/2 phase shift of m(t)). If we will add the two balanced modulator signal we will get the LSB and if we will subtracts the lower modulator o/p from the upper modulator output them we will get the USB.

Sin(wct)

Cos(wct)

Balance Modulator

Balance Modulator

900 Phase shift (HT)

Adder / Substracter

900 Phase shift (HT)

Carrier Oscillator

m(t)