Music 171: Amplitude Modulation

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1 Music 7: Amplitude Modulation Tamara Smyth, Department of Music, University of California, San Diego (UCSD) February 7, 9

2 Adding Sinusoids Recall that adding sinusoids of the same frequency produces another sinusoid at that frequency. at different frequencies produces a signal that is no longer sinusoidal. at frequencies that are integer multiples of a fundamental f, produces a signal with a period of /f. amplitude (offset) time (s) magnitude frequency (Hz) Figure : Adding sinusoids at 5,, 5 Hz in both time and frequency domain. Music 7: Amplitude Modulation

3 Sinusoidal components that are integer multiples of a fundamental within an audio range are called harmonics. Music 7: Amplitude Modulation 3

4 A Note on Pitch and Frequency Generally, harmonic sounds are those for which we hear a pitch. A common pitch notation designates a pitch with an octave: C4 is middle C. A4 or A44 (44 Hz) is often used as a reference. Tones are often compared by the musical interval separating them (e.g. octave, perfect fifth, etc): There is a nonlinear relationship between pitch perception and frequency (an octave corresponds to a frequency ratio of :, a greater frequency change at higher registers). In equal-tempered tuning, there are evenly spaced tones (semitones) in an octave, The frequency n semitones above A44 is 44 n/ Hz. Music 7: Amplitude Modulation 4

5 The frequency n semitones below A44 is 44 n/ Hz. Music 7: Amplitude Modulation 5

6 Summing Sinusoids Close in Frequency What happens when we two sinusoids having frequencies that are not harmonically related? Consider the sum of two sinusoids very close in frequency: 3.5 amplitude time (s) Figure : Sinusoids at 8 and Hz. Music 7: Amplitude Modulation 6

7 6 5 4 amplitude time (s) Figure 3: Sinusoids at 8 and Hz amplitude time (s) Figure 4: Sinusoids at 8 and Hz. Music 7: Amplitude Modulation 7

8 Beat Notes If we zoom out further, we can see that it is a sinusoid with a low-frequency sinusoidal amplitude envelope. amplitude time (s) Figure 5: Sinusoids at 8 and Hz. An audio version of this note (with frequencies at 8 and Hz) can be heard here. The beating comes about by adding two sinusoids that are very close in frequency. What is going on? Music 7: Amplitude Modulation 8

9 Multiplication of Sinusoids Recall, that to apply an envelope on a signal, the envelope is multiplied by the signal. Adding two sinusoids close in frequency is the same as multiplying a low-frequency sinusoid with one that is higher-frequency. This can be shown mathematically to be precisely true! Cosine Product formula, cos(a)cos(b) = cos(a+b)+cos(a b), we can show that x(t) = cos(π()t)cos(π()t) = [cos(π()t)+cos(π(8)t)]. Music 7: Amplitude Modulation 9

10 Beat Spectrum Sinusoidal multiplication can therefore be expressed as addition (which makes sense because a signal s spectrum can be represented by the sum of sinusoids) Figure 6: Beat note waveform and spectrum made by adding sinusoids at frequencies 8 Hz and Hz. Spectral components are not those of the multiplied sinusoids (at and Hz), but their sum ( Hz) and difference (8 Hz). Music 7: Amplitude Modulation

11 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 though we will not discuss the last one here. Music 7: Amplitude Modulation

12 Ring Modulation Ring modulation (RM) (e.g. beat note): modulator is applied directly to the amplitude of the carrier: x(t) = cos(πf t)cos(πf c t). Results in the sum of sinusoids: x(t) = cos(π(f c f )t)+ cos(π(f c+f )t), f f f c f f f c f frequency Figure 7: Spectrum of ring modulation. Neither carrier nor modulator are in the spectrum. Sometimes called double-sideband (DSB) modulation because of produced sidebands. Music 7: Amplitude Modulation

13 Double Sideband Modulation RM can be realized by multiplying any two signals together (not just oscillators). Total number of frequency components: N total = N N ( times the product of the number of components in each signal). Music 7: Amplitude Modulation 3

14 Classic Amplitude Modulation Classic amplitude modulation (AM) is more general. Modulating signal includes a constant (DC component): x(t) = (A +cos(πf t))cos(πf c t), (where the first term is the modulating signal.) DC component makes the modulating signal unipolar, i.e., the entire signal is greater than zero. Unipolar signal 3 Amplitude Time (s) Figure 8: A unipolar signal. Music 7: Amplitude Modulation 4

15 Effects of the DC component Multiplying out the above equation, we obtain x(t) = A cos(πf c t)+cos(πf t)cos(πf c t). The carrier frequency is now present in the spectrum. The second term can be expanded in the same way as was done for RM (i.e. the sidebands are identical). A A f f f c f f f c f frequency Figure 9: Spectrum of amplitude modulation. Music 7: Amplitude Modulation 5

16 RM and AM Spectra Sidebands are identical, but AM has center frequency f c in the spectrum. A A f f f c f f f c f frequency Figure : Spectrum of amplitude modulation. f f f c f f f c f frequency Figure : Spectrum of ring modulation. A DC offset in the modulator results in a spectrum with the carrier frequency f c, at an amplitude equal to A. Music 7: Amplitude Modulation 6

17 RM and AM waveforms Waveforms for AM and RM showing effect of DC offset in the modulator: 3 Amplitude Modulation Amplitude Time (s) Ring Modulation Amplitude Time (s) Figure : Amplitude and ring modulation. Music 7: Amplitude Modulation 7

CMPT 368: Lecture 4 Amplitude Modulation (AM) Synthesis

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