Convolution and modulation theorems
Convolving in one domain is multiplying in the other. The first theorem lets you analyze filters; the second lets you move signals in frequency, and all of radio comes from it.
01Two theorems that are the same one
They have already appeared separately; it pays to see them together because their symmetry is what organizes half of signal processing:
The first is the one that makes it possible to analyze filters. The second is the one that makes it possible to move signals in frequency, and from it come radio, television, the cell phone and anything that transmits.
| If in time… | In frequency… | Application |
|---|---|---|
| It is convolved with \( h \) | It is multiplied by \( H \) | Filtering |
| It is multiplied by a cosine | The spectrum splits in two and shifts | Modulating |
| It is multiplied by a window | The spectrum widens | Spectral analysis |
| It is multiplied by an impulse train | The spectrum repeats | Sampling |
The last two rows have already been used without naming them: spectral leakage and the replicas from sampling are the same theorem applied to two different multiplications.
02Amplitude modulation
The most direct case. Multiplying a signal by a carrier shifts its entire spectrum up to the carrier frequency, and two copies appear: the sidebands. If the carrier is also added, the original signal is drawn in the envelope, and it can be recovered with a diode and a capacitor.
On top the modulated signal with its envelope; below the spectrum. Move the modulation index and watch both: how much power goes into the carrier and what happens when it goes past 1.
03The variants, and what each one gains
| Type | Bandwidth | Efficiency | Demodulation |
|---|---|---|---|
| AM with carrier | \( 2B \) | 33 % at most | A diode: envelope detector |
| DSB without carrier | \( 2B \) | 100 % | Coherent: the carrier is needed at the receiver |
| Single sideband | \( B \) | 100 % | Coherent, and also sensitive to frequency errors |
The first row has the worst efficiency and yet it was the one used for eighty years in commercial radio. The reason is economic and lies on the receiver side: transmitting the carrier wastes power at the transmitter, but it allows the receiver to be a diode, a capacitor and a resistor. With millions of receivers and a single transmitter, it pays to spend on the transmitter.
With a sinusoidal modulating signal, the total power is \( P_c(1 + m^2/2) \) and the power carried by the sidebands —the only power that carries information— is \( P_c\,m^2/2 \). The ratio, in the best case \( m = 1 \), gives \( 0.5/1.5 = 33.3\,\% \). Two thirds of the power of an AM transmitter go into a carrier that says nothing. The lab calculates that number for each \( m \).
04Going past one
With \( m \gt 1 \) the envelope crosses zero and flips over. The spectrum is still fine —the information was not lost— but the envelope detector, which only follows the upper contour, delivers a distorted signal.
That explains an asymmetry that is confusing at first: overmodulation does not break the signal, it breaks the cheap receiver. A coherent detector would recover the modulating signal without any problem. That is why AM transmitters include a limiter that prevents going past \( m = 1 \), and why in DSB without carrier the concept of modulation index does not even apply.
05In the lab
Generate an AM signal with an analog multiplier and measure the maximum and minimum values of the envelope on the oscilloscope. Calculate \( m = (A_{\max} - A_{\min})/(A_{\max} + A_{\min}) \) and compare it with the programmed value.
Modulate a 1 MHz carrier with a 5 kHz tone and measure the spectrum. Verify that the sidebands are at \( \pm 5 \) kHz and that their amplitude is \( m/2 \) relative to the carrier.
Raise the index above 1 and compare the output of an envelope detector with that of a coherent detector. Explain why one distorts and the other does not.
06Common mistakes
- Forgetting the \( 1/2\pi \) in the multiplication theorem, or carrying it along when working in hertz.
- Believing that overmodulation destroys the information. The spectrum remains complete; what fails is the envelope detector.
- Assuming that AM occupies the width of the modulating signal. It occupies twice that: there are two sidebands.
- Thinking that the carrier carries information. It carries none, and it takes two thirds of the power.
- Demodulating DSB without carrier with a diode. A coherent detector is needed.
07Self-assessment
What does the multiplication theorem say?
That multiplying in time is equivalent to convolving the spectra, divided by \( 2\pi \).
How much bandwidth does an AM signal of a 5 kHz signal occupy?
10 kHz: two sidebands of 5 kHz around the carrier.
What is the maximum efficiency of AM with carrier?
33.3 %, with \( m = 1 \). The other two thirds go into the carrier.
What happens with \( m \gt 1 \)?
The envelope flips over and the envelope detector distorts. The spectrum, on the other hand, is still fine.
Why was the carrier transmitted if it wastes power?
So that the receiver could be a diode with a capacitor. With millions of receivers, it pays to spend on the transmitter.
How is sampling related to this theorem?
Sampling is multiplying by an impulse train, and that convolves the spectrum with another train: hence the replicas.
08Further reading
- Leon W. Couch. Digital and Analog Communication Systems. 7th ed. (Spanish edition, “Sistemas de comunicación digitales y analógicos”), Pearson Prentice Hall, 2007. AM, DSB and SSB with their spectra, efficiencies and demodulation circuits.
- Alan V. Oppenheim and Alan S. Willsky. Signals and Systems. 2nd ed. (Spanish edition, “Señales y sistemas”), Prentice Hall, 1997. Chapter 8, with modulation treated as a property of the transform.
- Herbert L. Krauss, Charles W. Bostian and Frederick H. Raab. Solid State Radio Engineering. Wiley, 1980 (Spanish edition, “Estado sólido en ingeniería de radiocomunicación”, Limusa, 1984). Real modulator and demodulator circuits, with their limitations.