368: Sinusoids
CMPT Lecture 3 Sinusoids
Tamara Smyth, [e-mail] School of Computing Science, Simon Fraser University January 17, 2007
Sinusoids is a collective term referring to both sine and cosine functions. A sinusoid is a function having the following form: x(t) = A sin(t + ), where x is the quantity which varies over time and A f t t +
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peak amplitude radian frequency (rad/sec) = 2f frequency (Hz) time (seconds) initial phase (radians) instantaneous phase (radians)
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Figure 1: Sinusoid where A = 2, = 25, and = /4.
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Amplitude and Magnitude.
Period
peak amplitude: the nonnegative value of the waveforms peak (either positive or negative); often shortened to simply amplitude. instantaneous amplitude of x: the value of x(t) (either positive or negative) at time t. instantaneous magnitude: a nonnegative value given by |x(t)|; often shortened to simply magnitude.
The period T of a sinusoid is the time (in seconds) it takes to complete one cycle.
One cycle of a sinusoid is 2 radians.
/2
3/2
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Figure 2: Sinusoid.
Since sinusoids are periodic with period 2, an initial phase of is indistinguishable from an initial phase of 2. We may therefore restrict the range of so that it does not exceed 2. Typically we choose the range but we many also encounter < < , 0 < < 2.
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Phase
The initial phase , given in radians, tells us the position of the waveform cycle at t = 0. Also sometimes called: phase oset, phase shift, or phase factor.
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Frequency
The frequency f of the waveform is given in cycles per second or Hertz (Hz). Frequency is equivalent to the inverse of the period T of the waveform, f = 1/T Hz.
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The radian frequency , given in radians per second, is equivalent to the frequency in Hertz scaled by 2, = 2f (rad/sec).
Figure 3: Sine function = 0.
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Figure 4: Sine function with = /2.
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Sine and cosine functions.
The sine and cosine function are very closely related and can be made equivalent simply by adjusting their initial phase: sin = cos( ) or cos = sin( + ). 2 2
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Time-shifting a signal.
If a signal can be expressed in the form x(t) = s(t t1), we say x(t) is a time-shifted version of s(t).
s(t) x(t) = s(t2)
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y(t) = s(t+1)
Amplitude
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Figure 6: Time-shifting a signal.
1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (s)
Figure 5: Phase relationship between cosine and sine functions.
Consider the simple function s(t) = t 0 t 1. 0t21 2 t 3, Shifting the function by t1 = 2 seconds yields x(t) = s(t 2) = t 2 = t2
In calculus, the sine and cosine functions are derivatives of one other. That is, d sin d cos = cos and = sin . dt dt
which is simply s(t) with its origin shifted to the right, or delayed, by 2 seconds.
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Shifting the function by t1 = 1 yields seconds y(t) = s(t + 1) = t + 1 = t+1 0t+11 1 t 0,
Sinusoidal and Circular Motion
Consider a vector of length one (1), rotating at a steady speed in a plane, the vector tracing a circle with a radius equal to its length.
/2
which is simply s(t) with its origin shifted to the left, or advanced in time, by 1 seconds.
/4
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Figure 7: A vector rotating along the unit circle.
Each time the vector completes one rotation of the circle, it has completed a cycle of 2. The rate at which the vector completes one cycle is given by its frequency. The length of the vector is given by its amplitude (which for simplicity, in this case, is one (1)).
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Sinusoidal and Circular Motion cont.
The x- and y-axis are the horizontal and vertical lines intersecting at the circles centre.
y-axis (0, 1)
1 1 ( 2 , 2 )
Sinusoids and Circular Motion cont.
Projecting onto the x- and y-axis gives a sequence of points that resemble a cosine and sine function respectively.
amplitude >
(0, -1)
Figure 8: The vector coordinates are determined by projecting onto the x and y-axis.
Projecting the vector onto the x- and y-axes allows us to determine its coordinates in the xy-plane. If the vector is rotated in a counterclockwise direction, at angle from the positive x-axis, projecting onto both the x- and y-axes creates right angle triangles. Trigonometric identities, with knowledge of and the vector length, will help us determine the coordinates.
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(-1, 0)
! !
(1, 0) x-axis
time >
amplitude > time >
Figure 9: Projecting onto the x and y axis.
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
A positive phase indicates a shift to the left whereas a negative phase indicates a shift to the right.
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Adding two sinusoids of the same frequency
Adding two sinusoids of the same frequency but with ...