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Signal Spectra, Signal Processing
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Sinusoidal Signals
Fundamentals building blocks for
describing arbitrary signals.
General signals can be expressed as
sums of sinusoids (Fourier Theory)
Bridge to frequency domain.
Sinusoids are special signals to linear
filters (eigenfunctions)
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Time and Frequency
Closely related via sinusoids
Provide two different perspectives on
signals.
Many operations are easier to understand
in frequency domain.
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Sampling
Conversion from continuous time to
discrete time.
Required for DSP
Converts a signal to a sequence of
numbers (samples).
Straightforward operation.
With a few strange effects.
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Filtering
A simple, but powerful, class of operations
on signals
Filtering transforms an input signal into a
more suitable output signal .
Often best understood in frequency
domain.
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Introduction to Sinusoids
Formula for Sinusoidal Signals
The general formula for a sinusoidal signal is
A, f, and are parameters that characterize the sinusoidalsignal.
Where: A Amplitude: determines the height
of the sinusoid.
f
Frequency: determines the numberof cycles per second.
Phase: determines the location
of the sinusoid.
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Cosine Function
The properties of sinusoidal signals stem from theproperties of the cosine function:
Periodicity: cos(x+2) = cos(x)
Eveness: cos(-x) = cos(x)
Ones of cosine: cos(2k) = 1, for all integers k. Minus ones of cosine: cos((2k+1)) = -1, for all integers
k.
Zeros of cosine: cos(/2(2k+1)) = 0, for all integers k.
Relationship to sine function:sin(x) = cos(x /2) (shift to the right)
cos(x) = sin(x + /2) (shift to the left)
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Amplitude
The amplitude A is a scaling factor.
It determines how large the signal is
Specifically, the sinusoid oscillates
between +A and
A.
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Frequency and Period
Sinusoids are periodic signals
The frequency f indicates how many times
the sinusoid repeats per second
The duration of each cycle is called the
period of the sinusoid and denoted by T.
The relationship between frequency and
period is f = 1/T and T = 1/f
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Phase and Delay
The phase,, causes a sinusoid to be shiftedsideways
A sinusoid with phase, = 0, has a maximum at t =
0. A sinusoid that has a maximum at t = t1 can be
written as x(t) = Acos(2f(t t1))
Expanding the argument of the cosine leads tox(t) = Acos(2ft
2ft1)
Comparing to the general formula for a sinusoidreveals = -2ft1 and t1 = -/2 f
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Phase and Delay
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Example 1
Convert sine function to cosine functionand vice versa.
(a) x1(t) = 8sin(40t + /8)
(b) x2(t) = 6cos(70t - /9)
(c) x3(t) = 5sin(20t + 3/2)
(d) x4(t) = 10cos(30t + /4)
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Solution
(a) x1(t) = 8cos(40t + /8 - /2)
= 8cos(40t - 3 /8)
(b) x2(t) = 6sin(70t - /9 + /2)
= 6sin(70t + 7/18)
(c) x3(t) = 5cos(20t + 3/2 - /2)
= 5cos(20t + )
(d) x4(t) = 10sin(30t + /4 + /2)
= 10sin(30t + 3/4)
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Example 2
Find the equation that will represent afunction, given in the figure.
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Solution
Amplitude = 2
Period = 0.22 0.02 = 0.42 0.22 = 0.2 = T;
f = 1/T = 1/0.2 = 5 Hz
Time delay = 0.02 = (-)2(5)(0.02) = (-)/5
x(t) = 2cos(2(5)t
0.02) = 2cos(10t/5)