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Next: Fourier transform of typical Up: handout3 Previous: Continuous Time Fourier Transform
Properties of Fourier Transform
The properties of the Fourier transform are summarized below. The properties of the Fourier expansion ofperiodic functions discussed above are special cases of those listed here. In the following, we assume
and .
Linearity
Time shift
Proof: Let , i.e., , we have
Frequency shift
Proof: Let , i.e., , we have
Time reversal
Proof:
Properties of Fourier Transform http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
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Replacing by , we get
Even and Odd Signals and Spectra
If the signal is an even (or odd) function of time, its spectrum is an even (or odd)
function of frequency:
and
Proof: If is even, then according to the time reversal property, we have
i.e., the spectrum is also even. Similarly, if is odd, we have
i.e., the spectrum is also odd.
Time and frequency scaling
Proof: Let , i.e., , where is a scaling factor, we have
Properties of Fourier Transform http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
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Note that when , time function is stretched, and is compressed; when
, is compressed and is stretched. This is a general feature of Fourier
transform, i.e., compressing one of the and will stretch the other and vice versa. In
particular, when , is stretched to approach a constant, and is
compressed with its value increased to approach an impulse; on the other hand, when ,
is compressed with its value increased to approach an impulse and is stretched to
approach a constant.
Complex Conjugation
Proof: Taking the complex conjugate of the inverse Fourier transform, we get
Replacing by we get the desired result:
We further consider two special cases:
If is real, then
i.e., the real part of the spectrum is even (with respect to frequency ), and the imaginary part isodd:
If is imaginary, then
Properties of Fourier Transform http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
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i.e., the real part of the spectrum is odd, and the imaginary part is even:
If the time signal is one of the four combinations shown in the table (real even, real odd,
imaginary even, and imaginary odd), then its spectrum is given in the corresponding table
entry:
if is real if is imaginary
even, odd odd, even
if is Even
and even , even , even
if is Odd
and odd , odd , odd
Note that if a real or imaginary part in the table is required to be both even and odd at the same time, ithas to be zero.
These properties are summarized below:
1 real even , odd
2 real and even real and even
3 real and odd imaginary and odd
4 imaginary odd , even
5 imaginary and even imaginary and even
6 imaginary and odd real and odd
As any signal can be expressed as the sum of its even and odd components, the first three items aboveindicate that the spectrum of the even part of a real signal is real and even, and the spectrum of the oddpart of the signal is imaginary and odd.
Properties of Fourier Transform http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
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Symmetry (or Duality)
Or in a more symmetric form:
Proof: As , we have
Letting , we get
Interchanging and we get:
or
In particular, if the signal is even:
then we have
For example, the spectrum of an even square wave is a sinc function, and the spectrum of a sincfunction is an even square wave.
Multiplication theorem
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Proof:
Parseval's equation
In the special case when , the above becomes the Parseval's equation (Antoine Parseval
1799):
where
is the energy density function representing how the signal's energy is distributed along the frequency
axes. The total energy contained in the signal is obtained by integrating over the entire
frequency axes.
The Parseval's equation indicates that the energy or information contained in the signal is reserved, i.e.,the signal is represented equivalently in either the time or frequency domain with no energy gained orlost.
Correlation
The cross-correlation of two real signals and is defined as
Specially, when , the above becomes the auto-correlation of signal
Assuming , we have and according to
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multiplication theorem, can be written as
i.e.,
that is, the auto-correlation and the energy density function of a signal are a Fourier transform
pair.
Convolution Theorems
The convolution theorem states that convolution in time domain corresponds to multiplication infrequency domain and vice versa:
Proof of (a):
Proof of (b):
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Time Derivative
Proof: Differentiating the inverse Fourier transform with respect to we get:
Repeating this process we get
Time Integration
First consider the Fourier transform of the following two signals:
According to the time derivative property above
we get
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and
Why do the two different functions have the same transform?
In general, any two function and with a constant difference have the same
derivative , and therefore they have the same transform according the above method. This
problem is obviously caused by the fact that the constant difference is lost in the derivativeoperation. To recover this constant difference in time domain, a delta function needs to be added in
frequency domain. Specifically, as function does not have DC component, its transform does
not contain a delta:
To find the transform of , consider
and
The added impulse term directly reflects the constant in time domain.
Now we show that the Fourier transform of a time integration is
Proof:
First consider the convolution of and :
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Due to the convolution theorem, we have
Frequency Derivative
Proof: We differentiate the Fourier transform of with respect to to get
i.e.,
Multiplying both sides by , we get
Repeating this process we get
next previous
Next: Fourier transform of typical Up: handout3 Previous: Continuous Time Fourier TransformRuye Wang 2009-07-05
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