Saturday, September 25, 2010

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 communication systems, thebaseband 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 radio band 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 interesting special 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 Figure 1 explains the modulation property concept and derives the equation for the modulation property.
Figure 1: [video] Modulation property concepts and derivation
Figure 1 (mod_am-math-modulation-property.html)
Suppose the source signal to be modulated contains only one spectral component, i.e., the source is a sinusoid. The screencast video of Figure 2 shows how to apply the modulation property to predict the spectrum of the modulated signal. Once you have studied the video, try the exercises below to ensure that you understand how to apply the property for a variety of different modulating frequencies.
Figure 2: [video] Determine the spectrum of a modulated sinusoid
Figure 2 (mod_am-math-modulation-pitched.html)
The time-domain signal x(t) is a sinusoid of amplitude 2A with corresponding frequency-domain spectrum as shown in Figure 3.
Figure 3: Spectrum of the signal x(t)
Figure 3 (mod_am-math-spectrum_x.png)
Suppose x(t) is modulated by a sinusoid of frequency fm . For each of the exercises below, draw the spectrum of the modulated signal y(t)=cos(2πfmt)×x(t), where the exercise problem statement indicates the modulation frequency.

EXERCISE 1

fm = f0/5

EXERCISE 2

fm = f0/2

EXERCISE 3

fm = f0

EXERCISE 4

fm = 1.5f0

EXERCISE 5

fm = 2f0
Did you notice something interesting when fm becomes larger than f0? 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 is similar 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 a sinusoid by a frequency greater than its own frequency folds the left-most component back into positive frequency.

Audio Demonstrations

The screencast video of Figure 4 demonstrates the aural effects of modulating a single spectral component, i.e., a sinusoid. The LabVIEW code for the demo is also described in detail, especially the use of an event structure contained in a while-loop structure (see video in Figure 5). The event structure provides an efficient way to run an algorithm with 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.
LabVIEW.png 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.
Figure 4: [video] Modulating a single sinusoid
Figure 4 (mod_am-math-demo1.html)
Figure 5: [video] LabVIEW implementation of AM demo using event structure
Figure 5 (mod_am-math-demo1-dgm.html)
The next screencast video (see Figure 6) demonstrates the aural effects of modulating two spectral components created by summing together a sinusoid at frequency f0 and another sinusoid 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 single pitch. If not, you perceive two distinct pitches.
LabVIEW.png 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.
Figure 6: [video] Modulating a pair of sinusoids
Figure 6 (mod_am-math-demo2.html)
The third demonstration (see Figure 7) illustrates the effect of modulating a music clip and a speech signal. You can obtain Interesting special effects because the original source spectrum simultaneously shifts to a higher and lower frequency.
LabVIEW.png 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).
Figure 7: [video] Modulating a music clip and a speech signal
Figure 7 (mod_am-math-demo3.html)

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