Concept 05 / 05Communication Systems
Amplitude Modulation: Modulation Index, Sidebands, Bandwidth, Power, Production and Detection
Deriving the AM waveform from first principles: modulation index from the envelope, the two sidebands and the bandwidth 2fm, the power split, how AM is generated and detected, and why AM is noisier than FM.
In amplitude modulation the amplitude of a high frequency carrier is made to follow the instantaneous value of the message, while the frequency and phase of the carrier are left strictly alone. The diagrams above show the three waveforms and the resulting spectrum, and everything asked in JEE Main on this topic follows from the few lines of algebra below.
Setting up the waveform.
Let the message be a single tone and the carrier , with . Amplitude modulation replaces the constant carrier amplitude by the time varying amplitude . Therefore
.
Taking outside the bracket,
, where .
The dimensionless number is the modulation index, also written as and quoted as a percentage. It measures how deeply the carrier amplitude is swung by the message. The quantity in brackets, , is the envelope: it is a faithful copy of the message sitting on a pedestal , and recovering the message is nothing more than tracing that envelope.
Modulation index from the picture. The envelope oscillates between and . Adding and subtracting, and , so
.
This is the form to use whenever a waveform or an oscilloscope trace is given, since and are the only things you can read off directly.
Overmodulation, and why must not exceed one. If then and the envelope tries to go negative. An amplitude cannot be negative; what physically happens is that the envelope collapses to zero, the carrier undergoes a sudden phase reversal of , and the observed envelope is the modulus of the intended one. The negative peaks of the message are chopped off, extra harmonics appear and the recovered audio is badly distorted, spilling out of its allotted channel and interfering with neighbours. This is why broadcasters keep below one, typically around . Note carefully that the distortion is a fault of envelope detection: the information in the sidebands is still intact, and a more elaborate coherent detector would recover it perfectly.
Where do the sidebands come from?
Expand the product in the modulated wave.
.
Use the identity on the second term:
.
This is the central result of the whole chapter. Although the picture in the time domain suggests one wave of wobbling amplitude, the AM wave is mathematically the sum of three pure sinusoids of constant amplitude:
• the carrier at , of amplitude ;
• the lower side band at , of amplitude ;
• the upper side band at , of amplitude .
The spectrum diagram above shows these three lines. A modulator is therefore best thought of as a frequency shifter: it takes the message at and moves it up so that it straddles the carrier.
Bandwidth. The highest frequency present is and the lowest is , so
.
If the message occupies a band of frequencies up to , every component produces its own pair of sidebands and the AM wave occupies . This single fact governs the whole radio dial. Indian AM broadcast channels are spaced kHz apart, so the audio fed to an AM transmitter must be filtered to about kHz, which is precisely why AM radio sounds dull compared with FM: the top two octaves of the music have been thrown away before transmission.
Power in an AM wave.
Let the wave be delivered to a resistance (the radiation resistance of the antenna). The average power of a sinusoid of amplitude is . Hence
, and each side band carries .
Because the three components are at different frequencies, their average powers simply add with no cross terms, so the total transmitted power is
.
Since power is proportional to the square of the antenna current, the same statement in terms of current is .
The fraction of the transmitted power that carries information is
.
Even at the maximum permissible this is only . At least two thirds of the power of every AM station is spent radiating a bare carrier that conveys nothing at all, and at a realistic the figure is . That is the price paid for a receiver that needs only a diode. Point to point services, which can afford clever receivers, suppress the carrier (double side band suppressed carrier) or even one of the side bands (single side band), saving power and halving the bandwidth.
Several message frequencies at once. If tones with indices modulate the same carrier, their side band powers add, and since each is proportional to the square of its index, the effective index is . Notice that indices add in quadrature, never directly.
How an AM wave is produced.
The message and the carrier are added and the sum is passed through a device with a square law characteristic, such as a diode or a transistor biased on the curved part of its characteristic. Write the input as and the output as . Then
.
Squaring produces a dc term, terms at and from the squares, and the all important cross term , which is a pair of side bands at . So the output contains components at , , , and , plus dc.
A band pass filter centred at , of bandwidth , rejects everything except and . What survives is , a genuine AM wave with modulation index . It is then raised to the required level by a power amplifier and fed to the transmitting antenna.
How an AM wave is detected.
The receiving antenna intercepts a minute voltage, which is amplified, selected by a tuned circuit, and in a practical superheterodyne receiver mixed with a local oscillator so that every station is translated to a common intermediate frequency of kHz where the amplification and filtering can be done once and done well. The recovery of the message then happens in the detector.
The envelope detector is a diode followed by a parallel combination of and . The diode passes only the positive half cycles, so the output is a rectified AM wave. On each carrier peak the capacitor charges rapidly to that peak through the small forward resistance of the diode; between peaks the diode is reverse biased and the capacitor discharges slowly through . The voltage across therefore hops from peak to peak and traces the envelope. A series capacitor afterwards blocks the steady component, which corresponds to the carrier pedestal , leaving the message.
The design condition is . Both inequalities are essential and each failure has a name. If is too small the capacitor discharges appreciably between carrier cycles and the output has a large carrier frequency ripple riding on the audio. If is too large the capacitor cannot discharge fast enough to follow the envelope when the envelope is falling steeply, and the output becomes a straight downward slope instead of a curve, a defect called diagonal clipping or negative peak clipping. Since the envelope has a maximum fractional rate of fall determined by and , the precise condition for no diagonal clipping works out to , which also shows that deep modulation of a high audio frequency is the hardest case.
Why AM is more vulnerable to noise than FM.
Atmospheric and man made noise adds itself to the wave as a fluctuating voltage, which means it shows up as a fluctuation of amplitude. In AM the information is carried by the amplitude, so the detector cannot possibly tell a genuine amplitude variation from a noise spike: the crackle of a lightning stroke goes straight to the loudspeaker.
In FM the information is carried by the instantaneous frequency, and the amplitude is meant to be constant. The receiver can therefore place a limiter before the demodulator, a stage that deliberately clips the wave to a fixed amplitude and so removes the amplitude fluctuations, taking most of the noise with them and losing nothing, since the message was never in the amplitude. FM also trades bandwidth for quality: it deliberately occupies about kHz per channel against the kHz of AM, and a wider frequency deviation buys a proportionately better output signal to noise ratio, an exchange that has no counterpart in AM. Two further advantages are the capture effect, by which the stronger of two co-channel FM signals suppresses the weaker altogether, and pre-emphasis at the transmitter with matching de-emphasis at the receiver, which lifts the weak high audio frequencies above the noise before transmission. Finally, AM broadcasting occupies the medium wave band where atmospheric noise is severe, while FM lives at VHF where it is far weaker.
Key Takeaways and Formulae:
• AM wave: with , and .
• Expanded form: . Three frequencies, two side bands of amplitude each.
• Bandwidth , or for a message band. Channel spacing in AM broadcasting is kHz, so audio is limited to about kHz.
• Power: , side band fraction , maximum at . Antenna current .
• Several tones: , added in quadrature.
• Production: add message and carrier, pass through a square law device , then a band pass filter at of width ; the result has .
• Detection: rectifier plus envelope detector with ; too large an gives diagonal clipping, with the strict limit .
• Common mistakes: writing the bandwidth as or as ; forgetting that the carrier amplitude, not the frequency, varies; using ; adding modulation indices directly instead of in quadrature; believing overmodulation destroys the information rather than the envelope.
Brain-Teaser 1: Reading everything off one equation.
An amplitude modulated voltage is volts. Find the modulation index, the maximum and minimum envelope amplitudes, all the frequencies present, the bandwidth, the percentage of power in the side bands, and the total power if the carrier alone delivers W.
Solution: Comparing with gives V, , Hz kHz and Hz kHz.
The envelope swings between V and V. As a check, , which agrees.
Three frequencies are present: the lower side band at kHz, the carrier at kHz and the upper side band at kHz. Each side band has amplitude V. The bandwidth is kHz, a thousandth of the carrier frequency, which is why the modulated wave still looks like a slightly wobbly carrier.
For the power, , so the side band fraction is , that is . Only about a seventh of the radiated power carries the message. The total power is W, of which W is the informationless carrier and W goes into each side band.
Brain-Teaser 2: Three tones and a hidden overmodulation.
A carrier is modulated simultaneously by three tones with individual modulation indices , and . Is the transmitter overmodulated? What is the total power if the unmodulated carrier power is kW? And why do the indices combine as they do?
Solution: The effective index is .
Since exceeds one the transmitter is overmodulated, even though no individual tone comes close to the limit and a careless engineer watching the tones one at a time would see nothing wrong. Note also the deliberate trap: and alone give exactly , sitting precisely on the boundary, so the third tone, the smallest of the three, is what pushes the system over.
The total power is W.
Why quadrature? Each tone creates its own pair of side bands at distinct frequencies , and . Sinusoids at different frequencies are orthogonal over a long time, so their average powers add with no interference terms. The side band power of the th tone is , so the total side band power is . Defining an equivalent single tone index by gives at once. Powers add, amplitudes do not, and that is the entire origin of the square root.
Brain-Teaser 3: Choosing the detector time constant.
An envelope detector is to demodulate a carrier of MHz carrying audio up to kHz at a modulation index of . Find the permissible range of and explain precisely what goes wrong at each end.
Solution: The carrier period is s and the shortest audio period is s, so the crude requirement already places somewhere in the region of tens of microseconds.
The strict upper limit comes from diagonal clipping. The envelope is and the capacitor, once the diode stops conducting, decays as , whose initial rate of fall is . Tracking requires the capacitor to fall at least as fast as the envelope at every instant, that is . Minimising the ratio over a cycle gives the standard condition
.
Substituting and rad/s: and , so s s.
A sensible design therefore uses of about to s, comfortably above s and comfortably below s.
What fails at each end. If is too small, say s, the capacitor loses a large part of its charge between successive carrier peaks, so the output is a sawtooth at MHz rather than a smooth envelope. The audio is still there, but a strong carrier frequency ripple accompanies it and must be filtered out later, wasting signal.
If is too large, say s, the capacitor simply cannot follow the envelope on its steep downward stretches. The diode stays reverse biased for many carrier cycles while the true envelope plunges away beneath the slowly decaying capacitor voltage, and the output shows a straight diagonal line where a curve should be. The recovered audio is distorted, with the negative peaks flattened, and the distortion is worst at high audio frequencies and deep modulation, which is exactly what the formula says: both a large and a large shrink the permitted .
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