Sinusoids
CMPT 318: Lecture 3
Sinusoids
• “Sinusoids” is a collective term referring to both sine
and cosine functions.
Tamara Smyth, tamaras@cs.sfu.ca
School of Computing Science,
Simon Fraser University
• A sinusoid is a function of time having the following
form:
x(t) = A sin(ωt + φ),
where x is the quantity which varies over time and
January 16, 2006
A
ω
f
t
φ
ωt + φ
,
,
,
,
,
,
peak amplitude
radian frequency (rad/sec) = 2πf
frequency (Hz)
time (seconds)
initial phase (radians)
instantaneous phase (radians)
Sinusoidal Signal
2
Amplitude
1
0
−1
−2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Figure 1: Sinusoid where A = 2, ω = 2π5, and φ = π/4.
1
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
Amplitude and Magnitude.
2
Phase and Period.
• The term peak amplitude, often shortened to
amplitude, is the nonnegative value of the
waveform’s peak (either positive or negative).
• The initial phase φ, given in radians, tells us the
position of the waveform cycle at t = 0. Also
sometimes called:
– phase offset
– phase shift
– phase factor
• The instantaneous amplitude of x is the value of
x(t) (either positive or negative) at time t.
• The instantaneous magnitude, or simply
magnitude, of x is nonnegative and is given by
|x(t)|.
• One cycle of a sinusoid is 2π radians.
• The period T of a sinusoid is the time (in seconds) it
takes to complete one cycle.
• Since sinusoids are periodic with period 2π, an initial
phase of φ is indistinguishable from an initial phase of
φ ± 2π.
– We may therefore restrict the range of φ so that it
does not exceed 2π.
– Typically we choose the range
−π < φ < π,
but we many also encouter
0 < φ < 2π.
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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Frequency
Sine and cosine functions.
• 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,
• The sine and cosine function are very closely related
and can be made equivalent simply by adjusting their
initial phase:
π
π
sin θ = cos(θ − )
or
cos θ = sin(θ + ).
2
2
1
Hz.
Amplitude
f = 1/T
• The radian frequency ω, given in radians per second,
is equivalent to the frequency in Hertz scaled by 2π,
ω = 2πf
0.5
0
−0.5
−1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
(rad/sec).
Figure 2: Phase relationship between cosine and sine functions.
• In calculus, the sine and cosine functions are
derivatives of one other. That is,
d sin θ
d cos θ
= cos θ
and
= − sin θ.
dt
dt
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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y(t) = s(t + 1) = t + 1
= t+1
• If a signal can be expressed in the form
we say x(t) is a time-shifted version of s(t).
1
y(t) = s(t+1)
1
0 1
A positive phase indicates a shift to
the left whereas a negative phase indicates a shift to the right.
1
0 1 2 3
0≤t+1≤1
− 1 ≤ t ≤ 0,
which is simply s(t) with its origin shifted to the left,
or advanced in time, by 1 seconds.
x(t) = s(t − t1),
x(t) = s(t−2)
6
• Shifting the function by t1 = −1 seconds yields
Time-shifting a signal.
s(t)
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
−1 0 1 2
Figure 3: Time-shifting a signal.
• Consider the simple function
s(t) = t
0 ≤ t ≤ 1.
• Shifting the function by t1 = 2 seconds yields
x(t) = s(t − 2) = t − 2
= t−2
0≤t−2≤1
2 ≤ t ≤ 3,
which is simply s(t) with its origin shifted to the
right, or delayed, by 2 seconds.
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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Sinusoidal and Circular Motion
Sinusoidal and Circular Motion cont.
• 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.
• The x- and y-axis are the horizontal and vertical lines
intersecting at the circle’s centre.
y-axis
(0, 1)
π/2
π/4
( √12 , √12 )
1
π
θ
(-1, 0)
(1, 0)
x-axis
2π
(0, -1)
3π/2
Figure 5: The vector coordinates are determined by projecting onto the x and y-axis.
Figure 4: A vector rotating along the unit circle.
• Projecting the vector onto the x- and y-axes allows us
to determine its coordinates in the xy-plane.
• Each time the vector completes one rotation of the
circle, it has completed a cycle of 2π.
• The rate at which the vector completes one cycle is
given by its frequency.
• If the vector is rotated in a counterclockwise direction,
at angle θ from the positive x-axis, projecting onto
both the x- and y-axes creates right angle triangles.
• The length of the vector is given by its amplitude
(which for simplicity, in this case, is one (1)).
• Trigonometric identities, with knowledge of θ and the
vector length, will help us determine the coordinates.
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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CMPT 318: Fundamentals of Computerized Sound: Lecture 3
Sinusoids and Circular Motion cont.
Adding two sinusoids of the same
frequency
• Projecting onto the x- and y-axis gives a sequence of
points that resemble a cosine and sine function
respectively.
• Adding two sinusoids of the same frequency but with
possibly different amplitudes and phases,
produces another sinusoid at that frequency.
3
sin(4πt)
2sin(4πt + π/4)
sum
2
Amplitude
amplitude −−−>
10
time −−−>
1
0
−1
−2
−3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Figure 7: Adding two sinusoids of the same frequency.
amplitude −−−>
time −−−>
• Recalling the expression for sinusoidal motion and
trigonometric identities1 , we may see that
x = A sin(ω0t + φ) = A sin(φ + ω0t)
= [A sin φ] cos ω0t + [A cos φ] sin ω0t
= B cos ω0t + C sin ω0t,
Figure 6: Projecting onto the x and y axis.
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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where the amplitude A is given by
√
A = B 2 + C 2,
Vector Addition
• Since one vector represents one sinusoid, to add two
sinusoids of the same frequency, we need only perform
vector addition.
and the phase φ is given by
y−axis
−1 C
.
φ = tan
B
V
Every sinusoid can be expressed as the sum
of a sine and cosine function, or equivalently,
an “in-phase” and “phase-quadrature” component.
U
U
V
x−axis
Figure 8: Adding sinusoids using vector addition.
• Since the vectors have the same frequency, they will
rotate as a unit and their sum will have the same
frequency.
• The sum vector u + v in Figure 8 also has its own x
and y component (from projecting onto the x- and
y-axes) and therefore may have a different amplitude
and phase.
CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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CMPT 318: Fundamentals of Computerized Sound: Lecture 3
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