Download Exercise – Acoustic Phonetics 433/633 Complex waves and spectra

Exercise – Acoustic Phonetics 433/633
Complex waves and spectra
1. Adding two simple waves (sine waves) is easy. It is the same as adding any two functions
f and g. The function f + g is drawn by plotting the values of f (x) + g(x) for all x in
the graph. In the figures, it is OK just to consider the integer values for x.
(a) Draw the graph of f + g in the figure below.
5
f
3
f+g
1
-1
-3
g
-5
0
2
4
6
8
Figure 1: 2 sine waves called A and B
10
x
(b) Draw the sum of waves of A and B in the figure below.
5
A
Pressure 0
A+B
B
-5
0
4
8
12
time (ms)
Figure 2: 2 sine waves called A and B
Fourier’s theorem says any complex wave is the sum of many—
possibly infinitely many—simple waves.
2. (a) What is the frequency (in Hz) of sine wave A?
Solution: 50Hz. There is 1/2 cycle in 10 milleseconds. How many cycles are
in 1 second?
1/2cycle
x
=
10ms
1s
1cycle
x
=
20ms
1000ms
1000 = 20x
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50 = x
(b) What is the frequency (in Hz) of sine wave B?
Solution: 250 Hz. There is 1 cycle in 4 milliseconds. How many cycles in
1,000 milliseconds?
1cycles
x
=
4ms
1000ms
1000 = 4x
250 = x
(c) What is the amplitude of sine wave A?
(c)
3
(d)
2
(d) What is the amplitude of sine wave B?
(e) On the graph provided, draw the spectrum of the complex wave A+B.
5
A
Amplitude
B
1
0Hz
250Hz
Frequency
(f) What is the fundamental frequency of the complex wave A+B?
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500Hz
Solution: 50 Hz. The fundamental frequency is the greatest common denominator (GCD) of the frequencies of the component waves. The GCD of of 50 and
200 is 50 because no larger number evenly divides both 50 and 200.
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