Frequency Domain Representation of Sinusoids: Continuous Time

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Frequency Domain Representation of Sinusoids
Continuous Time
Consider a sinusoid in continuous time
Frequency Domain Representation
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Example
Consider a sinusoid in continuous time
Represent it graphically as
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Continuous Time and Frequency Domain
In continuous time, there is a one to one
correspondence between a sinusoid and its
frequency domain representation
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Example
Let
msec
Given this sinusoid, its frequency, amplitude
and phase are unique
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Example
Consider a sinusoid in discrete time
Represent it graphically as
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Frequency Domain Representation of Sinusoids
Discrete Time
Same for a sinusoid in discrete time
Frequency Domain Representation
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Discrete Time and Frequency Domain
In discrete time there is ambiguity. All these
sinusoids have the same samples
with k integer
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Example
All these sinusoids have the same samples
and many more!!!
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Ambiguity in the Digital Frequency
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In Summary
A sinusoid with frequency
is indistinguishable from sinusoids with
frequencies
These frequencies are called aliases.
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Where are the Aliases?
Notice that, if the digital frequency is in the
interval
all its aliases are outside this interval
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Discrete Time and Frequency Domains
If we restrict the digital frequencies within the
interval
there is a one to one correspondence between
sampled sinusoids and frequency domain
representation (no aliases)
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Continuous Time to Discrete Time
Now see what happens when you sample a sinusoid
how do we relate analog and digital frequencies?

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Which Frequencies give Aliasing?
Aliases
k integer
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Example
Given a sinusoid with frequency
sampling frequency
the aliases (ie sinusoids with the same samples
as the one given) have frequencies
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Example
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Aliased Frequencies
aliases
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Sampling Theorem for Sinusoids
If you sample a sinusoid with frequency
such that , there is no
loss of information (ie you reconstruct the same
sinusoid)
magnitude
DAC
Digital to Analog Converter
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Extension to General Signals the Fourier Series
Any periodic signals with period can be
expanded in a sum of complex exponentials (the
Fourier Series) of the form
with
the fundamental frequency
The Fourier Coefficients
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Example
A sinusoid with period
We saw that we can write it in terms of complex
exponentials as
Which is a Fourier Series with
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Computation of Fourier Coefficients
For general signals we need a way of determining
an expression for the Fourier Coefficients. From
the Fourier Series multiply both sides by a
complex exponential and integrate
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Fourier Series and Fourier Coefficients
Fourier Series
Fourier Coefficients
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Example of Fourier Series
Period
Fundamental Frequency
Fourier Coefficients
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Plot the Coefficients
Fourier Coefficients
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Parsevals theorem
The Fourier Series coefficients are related to
the average power as
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Sampling Theorem
If a signal is a sum of sinusoids and B is the
maximum frequency (the Bandwidth) you can sample
it at a sampling frequency without
loss of information (ie you get the same signal
back)
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Example
it has two frequencies
The bandwidth is
The sampling frequency has to be
so that we can sample it without loss of
information
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Example
The bandwidth of a Hi Fidelity audio signal is
approximately since we cannot hear above this
frequency.
The music on the Compact Disk is sampled at
i.e. 44,100 samples for every second of music
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Example
For an audio signal of telephone quality we need
only the frequencies up to 4kHz. The sampling
frequency on digital phones is