Amplitude modulation (am) mathematics

Overview

Amplitude modulation ( AM ) is normally associated with communications systems; for example, you can find all sorts of "talk radio" stations on the AM band. In communicationsystems, the baseband signal has a bandwidth similar to that of speech or music (anywhere from 8 kHz to 20 kHz), and the modulating frequency is several orders of magnitude higher; the AM radioband is 540 kHz to 1600 kHz.

When applied to audio signals for music synthesis purposes, the modulating frequency is of the same order as the audio signals to be modulated. As described below, AM (also known as ring modulation ) splits a given signal spectrum in two, and shifts one version to a higher frequency and the other version to a lower frequency. The modulated signal is the sum of the frequency-shifted spectra, and can provide interestingspecial effects when applied to speech and music signals.

Modulation property of the fourier transform

The modulation property of the Fourier transform forms the basis of understanding how AM modifies the spectrum of a source signal. The screencast video of explains the modulation property concept and derives the equation for the modulation property.

Suppose the source signal to be modulated contains only one spectral component, i.e., the source is a sinusoid. The screencast video of shows how to apply the modulation property to predict the spectrum of the modulated signal. Once you have studied the video, try the exercises belowto ensure that you understand how to apply the property for a variety of different modulating frequencies.

The time-domain signal $x(t)$ is a sinusoid of amplitude $2A$ with corresponding frequency-domain spectrum as shown in .

Suppose $x(t)$ is modulated by a sinusoid of frequency $f_m$ . For each of the exercises below, draw the spectrum of the modulated signal $y(t)=\cos (2\pi f_{\mathrm{m}}t)\times x(t)$ , where the exercise problem statement indicates the modulation frequency.

Did you notice something interesting when $f_m$ becomes larger than $f_0$ ? The right-most negative frequency component shifts into the positive half of the spectrum, and the left-most positive frequency component shifts into the negative half of the spectrum. This effect issimilar to the idea of aliasing , in which a sinusoid whose frequency exceeds half the sampling frequency is said to be "folded back" into the principal alias. In the case of AM, modulating asinusoid by a frequency greater than its own frequency folds the left-most component back into positive frequency.

Audio demonstrations

The screencast video of demonstrates the aural effects of modulating a single spectral component, i.e., a sinusoid.The LabVIEW code for the demo isalso described in detail, especially the use of an event structure contained in a while-loop structure (see video in ). The event structure provides an efficient way to run an algorithmwith real-time interactive parameter control without polling the front panel controls. The event structure provides an alternative to the polled method described in Real-Time Audio Output in LabVIEW .

The LabVIEW VI demonstrated within the video is available here: am_demo1.vi . Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

The next screencast video (see ) demonstrates the aural effects of modulating two spectral components created by summing together a sinusoid at frequency f0 and anothersinusoid at frequency 2f0. You can obtain interesting effects depending on whether the spectral components end up in a harmonic relationship; if so, the components fuse together and you perceive a singlepitch. If not, you perceive two distinct pitches.

The LabVIEW VI demonstrated within the video is available here: am_demo2.vi . Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

The third demonstration (see ) illustrates the effect of modulating a music clip and a speech signal. You can obtain Interesting special effectsbecause the original source spectrum simultaneously shifts to a higher and lower frequency.

The LabVIEW VI demonstrated within the video is available here: am_demo3.vi . Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

The two audio clips used in the example are available here: flute.wav and speech.wav (speech clip courtesy of the Open Speech Repository, www.voiptroubleshooter.com/open_speech ; the sentences are two of the many phonetically-balanced Harvard Sentences , an important standard for the speech processing community).