The message rides on the carrier. Its shape is the envelope.
Ten minutes of reading. Move the sliders, then the code and the models of the lab will read like plain sentences.
Why modulate
Modulation changes a property of a high frequency carrier (its amplitude, frequency or phase) so that it follows a low frequency message. A voice signal of a few kilohertz cannot leave an antenna of a sensible size, and many stations sending their voices at once would all sit on top of each other. Moving each message to its own carrier frequency solves both problems: the antenna is short compared with the wavelength, and every station has its own slot in the spectrum.
The AM signal
Multiply out the brackets: Ac cos 2πfct + (mAc/2) cos 2π(fc − fm)t + (mAc/2) cos 2π(fc + fm)t. Three lines: the carrier, and a lower and an upper sideband, each of height m/2 times the carrier. In decibels the sidebands are 20 log10(2/m) below the carrier: 6.02 dB at m = 1 and 12.04 dB at m = 0.5. The block diagram in mathematical form is a summer (1 + m·x) followed by a multiplier with the carrier.
The modulation index
With m below 1 the envelope 1 + m cos 2πfmt stays above zero and is an exact copy of the message: a simple diode envelope detector gets the message back. At m = 1 (100 %) the envelope just touches zero. Above 1 the signal is over modulated: the envelope would go negative, the carrier phase flips by 180°, and the envelope detector returns a clipped, distorted message. Over modulation also adds extra lines and widens the spectrum, so the station spills into its neighbours. Move m in the demo above past 1 to see it.
DSB-SC and SSB
The carrier line of AM carries no information. Leave out the 1 and the product x·c is double sideband suppressed carrier: in Simulink, set the Constant of the AM model to 0. Both sidebands carry the same information, so one of them is enough: single sideband keeps one and halves the bandwidth. The phasing method of lab task 2 builds it from two products: for a tone, cos a cos b − sin a sin b = cos(a + b), the upper sideband alone.
The Spectrum Analyzer
The Spectrum Analyzer block (DSP System Toolbox) estimates the power spectrum of its input and plots it in dBm: decibels relative to 1 mW, into 1 ohm. A sine of amplitude A has a power of A²/2 W, so amplitude 1 reads 10 log10(500) = 26.99 dBm. The RBW (resolution bandwidth, 50 Hz in this lab) sets how close two lines may be and still look separate.
Peak Finder (in the Measurements menu of the analyzer) lists the highest peaks with their frequency and power, so the difference between the carrier and a sideband can be read directly.
In the virtual lab
In guided mode each step ticks itself when it is really done: the right file open, the right values set, the script or the simulation run, the value read and typed. Free mode leaves you alone with every file. The report keeps each figure with the values that made it.
What you should see
| File | Settings | Result |
|---|---|---|
| am_ssb | Fc = 10 Hz, a 1 Hz sine | DSB: lines at 9 and 11 Hz; SSB: one line at 9 Hz |
| sine_spectrum.slx | 1 kHz, amplitude 1 | one line at 1 kHz, 26.99 dBm |
| multiply.slx | x times x, km = 1 | output 0 to 1 V; lines at 0 and 2 kHz; with a Bias also 1 kHz |
| dsbsc.slx | 1 kHz times 10 kHz | lines at 9 and 11 kHz, no line at 10 kHz |
| am.slx | m = 0.5, fc = 5 kHz | carrier at 5 kHz, sidebands at 4 and 6 kHz, 12 dB lower |
| task1_am, task1_ssb | m = 1 and 0.5; SSB | sidebands of height 0.5 and 0.25; SSB one line at 4 kHz |
| task2_ssb.slx | phasing, signs +− | one line at 6 kHz; with ++ at 4 kHz |
| task3 (both) | 1 and 2 kHz on 10 kHz | 5 lines: 8, 9, 10, 11 and 12 kHz |
Check yourself
Q1Define modulation.
Modulation is the process of changing a property of a high frequency carrier (its amplitude, frequency or phase) in step with a low frequency message, so the message can be sent over a channel. It allows small antennas, sharing of the spectrum by many users (each on its own carrier) and a better match to the channel.
Q2What is amplitude modulation?
An analog modulation in which the amplitude of a fixed frequency carrier follows the message: s(t) = Ac[1 + m·x(t)] cos 2πfc t. The envelope of the carrier is a copy of the message. The spectrum has the carrier and two sidebands, so the bandwidth is twice the message bandwidth.
Q3Write the block diagram in mathematical form.
y(t) = [k + m(t)] · c(t). With a Constant k = 1, a message m(t) = m cos 2πfm t and a carrier c(t) = cos 2πfc t: the Add block makes 1 + m cos 2πfm t and the Product block multiplies it by the carrier, y = (1 + m cos 2πfm t) cos 2πfc t. With k = 0 the same diagram gives DSB-SC.
Q4Use a modulation frequency greater than the carrier frequency. Comment on the time and frequency domain.
In time the envelope no longer follows the message: the carrier makes less than one cycle per message cycle, so there is no smooth outline to follow, and an envelope detector cannot recover the message. In frequency the lower sideband fc − fm is negative and folds back to fm − fc, so the sidebands are no longer placed symmetrically around the carrier and may overlap the message band.
Q5Use a carrier frequency much greater than the modulation frequency. Comment on the time and frequency domain.
In time the carrier fills the envelope with many cycles and the envelope is a clean copy of the message (for m ≤ 1). In frequency the carrier line stands at fc with the two sidebands close to it at fc ± fm, well away from 0 Hz and from each other. This is the normal operating condition of AM.
Q6For the same case, draw the waveform for m = 0.5 and m = 1 and comment.
At m = 0.5 the envelope swings between 0.5 and 1.5 times the carrier amplitude and never reaches zero; the sidebands are 12 dB below the carrier. At m = 1 the envelope swings from 0 to 2 and touches zero once per message cycle; the sidebands are 6 dB below the carrier and carry the most power possible without distortion (one third of the total).
Q7What is over modulation, and how does it affect recovery of the message?
Over modulation is m > 1: the message peak is larger than the carrier, so 1 + m·x goes negative. The carrier phase reverses there and the envelope becomes |1 + m·x|, which is not the message. An envelope detector then gives a clipped, distorted message, and the extra harmonics widen the spectrum into the neighbouring channels.