CMPT 368: Lecture 4 Amplitude Modulation (AM) Synthesis Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 8, 008
Beat Notes What happens when we add two frequencies that are not harmonically related? The Beat note comes about by adding two sinusoids that are very close in frequency. Spectrum of Beat Note 0.9 0.8 0.7 0.6 Magnitude 0.5 0.4 0.3 0. 0. 0 05 0 5 0 5 30 35 Frequency (Hz) Figure : Beat Note made by adding sinusoids at frequencies 8 Hz and Hz. CMPT 368: Computer Music Theory and Sound Synthesis
The resulting waveform shows that there is a periodic, low frequency amplitude envelope superimposed on a higher frequency sinusoid. Beat Note Waveform (f0 = 0 Hz, f = Hz) 0.8 0.6 0.4 0. Amplitude 0 0. 0.4 0.6 0.8 0 0. 0.4 0.6 0.8..4.6.8 Time (s) Figure : Waveform of a Beat Note. What is going on? CMPT 368: Computer Music Theory and Sound Synthesis 3
Multiplication of Sinusoids What happens when we multiply a low frequency sinusoids with a higher frequency sinusoid? Begin by using the inverse Euler formula: x(t) = sin(π(0)t) cos(π()t) ( e jπ(0)t e jπ(0)t )( e jπ()t + e jπ()t ) = j = ] [e jπ()t e jπ()t + e jπ(8)t e jπ(8)t 4j = [sin(π()t) + sin(π(8)t)] which is a sum of real sine functions. From this we can now see that there will be four frequency components in the spectrum (including the negative frequencies), none of which are the two multiplied frequency components. Rather, the spectrum has their sum and the difference. Sinusoidal multiplication can therefore be expressed as an addition (which makes sense because all signals can can be represented by the sum of sinusoids). CMPT 368: Computer Music Theory and Sound Synthesis 4
Amplitude Modulation Modulation is the alteration of the amplitude, phase, or frequency of an oscillator in accordance with another signal. The oscillator being modulated is the carrier, and the altering signal is called the modulator. The spectral components generated by a modulated signal are called sidebands. There are three main techniques of amplitude modulation: Ring modulation Classical amplitude modulation Single-sideband modulation The latter is rarely used in computer music and will not be discussed here. CMPT 368: Computer Music Theory and Sound Synthesis 5
Ring Modulation Ring modulation (RM), introduced as the beat note waveform, occurs when modulation is applied directly to the amplitude input of the carrier modulator: x(t) = cos(πf t) cos(πf c t). Recall that this multiplication can also be expressed as the sum of sinusoids using the inverse of Euler s formula: x(t) = cos(πf t) + cos(πf t), where f = f c f and f = f c + f. f f f c f 0 f f c f frequency Figure 3: Spectrum of ring modulation. CMPT 368: Computer Music Theory and Sound Synthesis 6
Ring Modulation cont. Notice again in this type of modulation that neither the carrier frequency nor the modulation frequency are present in the spectrum. f f f c f 0 f f c f frequency Figure 4: Spectrum of ring modulation. Because of its spectrum, RM is also sometimes called double-sideband (DSB) modulation. Ring modulating can be realized without oscillators just by multiplying two signals together. The multiplication of two sounds produces a spectrum containing frequencies that are the sum and difference between each of the frequencies present in each of the sounds. CMPT 368: Computer Music Theory and Sound Synthesis 7
RM therefore produces components equal to two times the number of frequency components in one signal multiplied by the number of frequency components in the other. CMPT 368: Computer Music Theory and Sound Synthesis 8
Classic Amplitude Modulation Classic amplitude modulation (AM) is the more general of the two techniques. AM the modulating signal includes a constant or DC component in the modulating term, and is given by x(t) = [A 0 + cos(πf t)] cos(πf c t), (where the first term is the modulating signal.) Multiplying out the above equation, we obtain x(t) = A 0 cos(πf c t) + cos(πf t) cos(πf c t). The first term in the result above shows that the carrier frequency is actually present in the resulting spectrum. The second term can be expanded in the same way as ring modulation using the inverse Euler formula. CMPT 368: Computer Music Theory and Sound Synthesis 9
RM and AM Spectra Where the centre frequency f c was absent in RM, it is now present in classic AM. The sidebands are identical. A 0 A 0 f f f c f 0 f f c f frequency Figure 5: Spectrum of amplitude modulation. f f f c f 0 f f c f frequency Figure 6: Spectrum of ring modulation. A DC offset A 0 in the modulating term therefore has the effect of including the centre frequency f c at an amplitude equal to the offset. CMPT 368: Computer Music Theory and Sound Synthesis 0
Classic Amplitude Modulation Cont. Because of the DC component, the modulating signal is often unipolar, that is, the entire signal is above zero and the instantaneous amplitude is always positive. Unipolar signal 3 Amplitude 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) Figure 7: A unipolar signal. The signal without the DC offset that oscillates between the positive and negative is called bipolar CMPT 368: Computer Music Theory and Sound Synthesis
RM and AM waveforms The difference in waveforms between the amplitude and ring modulation (and therefore the effect of a DC offset in the modulating term) is shown signals is shown below. 3 Amplitude Modulation Amplitude 0 3 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) Ring Modulation Amplitude 0.5 0 0.5 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) Figure 8: Amplitude and ring modulation. CMPT 368: Computer Music Theory and Sound Synthesis