Continuous Fourier Transform
• We have introduced the continuous Fourier transform.
Forward
Backward
F  jw 



f t e
 jwt
dt
1
f t  
2

jwt


F
jw
e
dw


• The continuous Fourier transform defines completely and exactly the frequency domain, where the frequency domain is continuous and range un‐limited.
Examples of Fourier Transform Pairs
• Rectangular function (rectangular pulse signal)
• Derivation of its continuous F. T. (Sinc function)
Fourier transform of rectangular function
• Rectangular function can also be represented by the unit‐pulse function u(t) as T
T
u(t  )  u (t  )
2
2
where the unit‐step function is 1, t  0
u (t )  
0, t  0
• Hence, we have the Fourier transform pair:
A real‐valued function in frequency domain (sinc function)
Time and Frequency domains are dual
Fourier transform of right‐sided real‐exponential signal
A complex function in frequency domain
• Since
• The real and imaginary parts are
Impulse in Time and Frequency
• Derivation:
Impulse in Time and Frequency
• By the duality (or symmetry) between time and frequency domain
• Intuitive interpretation: the constant signal x(t)=1 for all t has only one frequency, namely DC, and we see that its transform is an impulse concentrated at =0.
Complex Exponential
• It says that a complex exponential signal of frequency 0 has a Fourier transform that is nonzero at only the frequency 0.
• Linear Property:
Sinusoids
• Derivation: Since
• By the linear property, we have
Periodic Signals
Represented as a Fourier series
• Example: time domain a periodic square wave
Frequency domain of squared wave
Basic Fourier Transform Pairs
Basic Fourier Transform Pairs
Properties of Fourier Transform Pairs
• Scaling Property
Properties of Fourier Transform Pairs
• Flip Property
• Derivation: from the scaling property, we have
Properties of Fourier Transform Pairs
• Time delay property
• Differentiation property
Basic Fourier Transform Properties
Basic Fourier Transform Properties
Symmetry Properties of Fourier Transform Pairs
• If we take complex conjugate of the spectrum, we obtain
• Hence, X*(‐jw) is the Fourier transform of x*(t)
Symmetry Properties of Fourier Transform Pairs
• Therefore, if x(t) is a real function, x(t)=x*(t), the above property reveals that X(jw) = X*(‐jw). Hence, we have
• Then, we can conduct that
Re{X(jw)} is an even function
Im{X(jw)} is an odd function
Symmetry Properties of Fourier Transform Pairs
• In other words, when x(t) is real, the real part of its Fourier transform X(jw) is even, and the imaginary part is odd.
• Similarly, we also have a symmetric property for magnitude (amplitude) and phase in polar form when x(t) is real
• That is, for a real‐valued signal, its the magnitude spectrum is an even function, and its phase spectrum is an odd function.
Example: magnitude and phase spectra of a real signal
Even function
Odd function
Multiplication in the frequency domain
• When multiplying two signals in the frequency domain, what will be obtained in the time domain?
• Convolution: a moving average operation:
• Convolution of two continuous‐time signals x(t) and h(t)
is defined as
y t  

 x ht   d

• It can be written in short by y  t   x (t )  h (t )
Illustration example of convolution
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݄ሺ‫ݐ‬ሻ
Convolution
• Convolution is commutative:
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ିஶ
• Let , then we have
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Animation example of convolution
when h(t) is even
Animation example of convolution
when h(t) is even
• Illustration of
convolution
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ିஶ
From Kuhn 2005 2D continuous convolution: optics example
Convolution in time domain
• Derivation
Convolution Property Derivation
• Interchange the order of integrals
• Let Convolution Property Derivation
• By substitution back,
• Convolution property is one of the most important properties in Fourier transform.
Convolution Property Concept
• Due to the duality of frequency and time domains, we also have the property that multiplication in the time domain corresponds to the convolution in the frequency domain:
• That is
Note that the above is only a conceptual formulation. For different definitions of the continuous Fourier transform, the convolution property is also a slight different according to how to decomposes the scaling 1/2.
Convolution Property
• For the continuous‐time Fourier transform pair defined below,
Forward
F  jw 



f t e  jwt dt
Backward

1
jwt


f t  
F
jw
e
dw

2 
• The convolution property is
Time domain Frequency domain
There is a scaling factor
Example: AM
Time domain multiplication
Frequency domain convolution
Finite Duration Signal and Band Limited Signal
• Finite Duration Signal: A signal x(t) is nonzero in [‐tb, tb] for some tb >0, and is zero elsewhere.
• Band Limited Signal: A signal’s frequency is nonzero in the frequency band [‐wb, wb] for some wb >0, and s zero elsewhere.
Finite duration in time domain
Band unlimited in frequency domain
Band limited in frequency domain
Infinite duration in time domain
Finite Duration Signal and Band Limited Signal
• Because a finite‐duration signal can be represented as the multiplication of some signal of a rectangular window.
• Time domain multiplication, frequency domain is the convolution of the sync function.
• Since the sync function is band unlimited. The convolution of it and any function is generally band unlimited.
Finite Duration Signal and Band Limited Signal
• Similarly, a band‐limited signal can be represented as the multiplication of some spectrum of a rectangular window in frequency domain.
• Frequency domain multiplication, time domain is the convolution with the sync function.
• Since the sync function is infinite durational. The convolution of it and any function is infinite durational.
Impulse Train
• Remember that the continuous Fourier transform of a periodic signal is an impulse sequence.
• What happens when the periodic signal is itself an impulse sequence?
• An example of such signal is the impulse train:
• According to the duality between the time and frequency domains of continuous Fourier transform, its Fourier transform should be an impulse sequence that is also periodic. Impulse Train
• What is the continuous Fourier Transform of p(t)?
Impulse Train
• Derivation: because p(t) is a periodic signal, it can be expressed by Fourier series:
where Impulse Train
• To determine the coefficients ak of Fourier series, we must evaluate the Fourier series integral over one period [Ts/2, Ts/2].
• Since p(t) is the impulse training with period Ts, there is only a single delta function, (t), within the period [Ts/2, Ts/2].
• Hence,
Impulse Train
• Remember that we have the property about the inner product of an impulse function and an arbitrary signal: • Hence, when and ି௝௞ఠೞ ௧
Impulse Train
• According to the transform pair of periodic signals, the continuous Fourier transform of a periodic signal is in general of the form of sum of delta functions centered at integer multiples of s:
• Hence, the Fourier transform of the impulse train p(t) is another impulse train