Continuous Time Fourier Transformation

1. Representation of non-cyclic signals: Continuous Time Fourier Transformation

In order to have a deeper understanding of the essence of Fourier transformation, we first start with the representation of Fourier series of a continuous time period square wave. That is, within a period

\ [X (t) = \ begin {cases} 1, & \ Text | T | <T_1 \ 0, & \ Text T_1 <| T | <t/2 \ end {cases} \]

Recurrence by period \ (T \), as shown in.

The Fourier series coefficient \ (A_k \) of the square wave signal is

\ [\ Tag {1} A_k = \ frac {2sin (k \ omega_0t_1)} {k \ omega_0t} \]
Formula \ (\ omega_0 = 2 \ PI/T \).

Another way to understand (1) is to treat it as a sample of an envelope function, that is

\ [\ Tag {2} ta_k = \ frac {2sin \ Omega T_1} {\ Omega} \ lvert _ {\ Omega = k \ omega_0} \]

This is,If \ (\ Omega \) is considered as a continuous variable, the $ {(2sin \ Omega t_1)}/{\ Omega} $ function represents the envelope of \ (ta_k, these coefficients are the samples obtained at the upper interval of the envelope.. If \ (T_1 \) is fixed, the envelope of \ (ta_k \) is independent of \ (T \), as shown in.

As shown in the figure, with the increase of \ (T \), the envelope is sampled at an increasingly intensive interval. As \ (T \) becomes arbitrary, the original periodic square wave approaches a rectangular pulse (that is, a non-periodic signal is retained in the time domain, it corresponds to a period of the original square wave ).

At the same time, Fourier series (multiplied by \ (T \) as samples on the envelope also become increasingly intensive. In a sense, with \ (T \ To \ infty \), the Fourier series approaches this envelope function.

This example illustrates the basic idea of building Fourier representation for non-cyclic signals,We can regard Non-cyclic signals as the limit of any large period..

Now let's consider a signal \ (X (t) \), which has a limited duration \ (2t_1 \), starting from this cycle signal, it can constitute a cycle signal \ (\ Tilde x (t) \), SO \ (X (t) \) is a cycle of \ (\ Tilde x (t. When \ (T \) is selected as a large value, \ (X (t) \) is connected to \ (\ Tilde x (t) \ for a longer period of time )\) consistent, and with \ (T \ To \ infty \), for any finite time value \ (T \), \ (\ Tilde x (t )\) it is equal to \ (X (t )\).

In this case, we consider representing \ (\ Tilde x (t) \) as Fourier series, select \ (-T/2 \ leqslant t \ leqslant T/2 \) as the integral range \).

\ [\ Tag {3} \ Tilde x (t) = \ sum _ {k =-\ infty} ^ {+ \ infty} a_ke ^ {JK \ omega_0t} \]

\ [\ Tag {4} A_k = \ frac {1} {t} \ int _ {-\ frac {t} {2 }}^ {\ frac {t} {2 }}\ Tilde x (t) e ^ {-JK \ omega_0t} DT \]

Formula \ (\ omega_0 = 2 \ PI/T \), due to \ (| T | <t/2 \), \ (\ Tilde x (t) = x (t) \), and in other places, \ (X (t) = 0 \), SO (4) can be rewritten

\ [\ Tag {5} A_k = \ frac {1} {t} \ int _ {-\ frac {t} {2 }}^ {\ frac {t} {2} x (t) e ^ {-JK \ omega_0t} dt = \ frac {1} {t} \ int _ {-\ infty} ^ {+ \ infty} x (t) e ^ {-JK \ omega_0t} DT \]

Therefore, define the envelope \ (x (J \ Omega) \) of \ (ta_k \)

\ [\ Tag {6} X (J \ Omega) =\int _ {-\ infty} ^ {+ \ infty} x (t) e ^ {-J \ omega t} DT \]

At this time, the coefficient \ (A_k \) can be written

\ [\ Tag {7} A_k = \ frac {1} {t} X (jk \ omega_0) \]

Combine (3) and (7), \ (\ Tilde x (t) \) can be expressed

\ [\ Tag {8} \ Tilde x (t) = \ sum _ {k =-\ infty} ^ {+ \ infty} \ frac {1} {t} X (jk \ omega_0) e ^ {JK \ omega_0t }=\ frac {1} {2 \ PI} \ sum _ {k =-\ infty} ^ {+ \ infty} X (jk \ omega_0) e ^ {JK \ omega_0t} \ omega_0 \]

With \ (T \ To \ infty \), \ (\ Tilde x (t) \) approaches \ (X (t) \), formula (8) the limit is changed to the \ (X (t) \) expression. In addition, when \ (T \ To \ infty \) has \ (\ omega_0 \ to 0 \), the right side of formula (8) is transitioned to a point.

Each item on the right can be regarded as the area of the rectangle with the height of \ (x (jk \ omega_0) e ^ {JK \ omega_0t} \) and the width of \ (\ omega_0. Formula (8) and formula (6) are changed

\ [\ Tag {9} \ boxed {x (t) = \ frac {1} {2 \ PI} \ int _ {-\ infty} ^ {+ \ infty} X (J \ Omega) e ^ {J \ omega t} d \ Omega} \]

\ [\ Tag {10} \ boxed {x (J \ Omega) =\int _ {-\ infty} ^ {+ \ infty} x (t) e ^ {-J \ omega t} DT} \]

(9) and (10) are calledFourier transform pair. Function \ (x (J \ Omega) \) is called \ (X (t) \)Fourier transform or Fourier Integral, Also knownSpectrumAnd (9) is calledFourier Inversion.

• Example 1
  

• Example 2
  
  

Sinc FunctionsThe common form is

\ [\ Tag {11} SiNc (\ theta) = \ frac {sin \ pi \ Theta} {\ pi \ Theta} \]

2. Fourier transformation of Periodic Signals

Consider a signal \ (X (t) \), and its Fourier transform \ (x (J \ Omega) \) is an area of \ (2 \ pi \), an independent impulse that appears at \ (\ Omega = \ omega_0 \), that is

\ [\ Tag {12} X (J \ Omega) = 2 \ pi \ delta (\ omega-\ omega_0) \]

To obtain the \ (X (t) \) corresponding to \ (x (J \ Omega) \), you can apply the inverse transformation formula of formula (9) to obtain

\ [\ Tag {13} x (t) = \ frac {1} {2 \ PI} \ int _ {-\ infty} ^ {+ \ infty} 2 \ pi \ delta (\ omega-\ omega_0) e ^ {J \ omega t} d \ Omega = e ^ {J \ omega_0 t} \]

Promote the above results. If \ (x (J \ Omega) \) is a linear combination of a group of Impulse functions at the same frequency interval, that is

\ [\ Tag {14} X (J \ Omega) = \ sum _ {k =-\ infty} ^ {+ \ infty} 2 \ PI A_k \ delta (\ omega-k \ omega_0) \]

Then formula 9 is available.

\ [\ Tag {15} x (t) = \ sum _ {k =-\ infty} ^ {+ \ infty} a_ke ^ {JK \ omega_0 t} \]

We can see that formula (15) represents the Fourier series given by a periodic signal. Therefore,The Fourier transformation of a periodic signal with a Fourier series coefficient of \ (\ {A_k \} \) can be seen as a series of Impulse functions appearing at the frequency of the harmonic relationship., The area of the Impulse Function on the \ (k \) subharmonic frequency \ (k \ omega_0 \) is \ (k \) the \ ({2 \ PI} \) times of the Fourier series coefficient \ (A_k.

3. Continuous Time Fourier Transformation

For convenience, we use the following symbol to represent the Fourier Transformations \ (X (t) \) and \ (x (J \ Omega) \).

\ [X (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (J \ Omega) \]

3.1. Linear

If

\ [X (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (J \ Omega) \]

And

\ [Y (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} y (J \ Omega) \]

Then

\ [\ Tag {16} \ boxed {ax (t) + by (t) \ overset {\ displaystyle {\ mathcal {f }}}{ \ leftrightarrow} ax (J \ Omega) + by (J \ Omega)} \]

3.2. time shifting

If

\ [X (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (J \ Omega) \]

Then

\ [\ Tag {17} \ boxed {x (t-t_0) \ overset {\ displaystyle {\ mathcal {f }}}{ \ leftrightarrow} e ^ {-J \ Omega T_0} X (J \ Omega)} \]

This property indicates that the signal shifts in time and does not change its Fourier transform modulus.

3.3. condensed and condensed Symmetry

If

\ [X (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (J \ Omega) \]

Then

\ [\ Tag {18} \ boxed {x ^ * (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} x ^ * (-J \ Omega)} \]

If \ (X (t) \) is a real function, then \ (x (J \ Omega) \) has a bounded symmetry, that is

\ [\ Tag {19} \ boxed {x (-J \ Omega) = x ^ * (J \ Omega) \ qquad [x (t) is real]} \]

That is to say,The real part of Fourier transformation is the occasional function of frequency, while the imaginary part is the odd function of frequency..

3.4. Differentiation and integration

\ [\ Tag {20} \ boxed {\ frac {dx (t )} {DT} \ overset {\ displaystyle {\ mathcal {f }}}{ \ leftrightarrow} J \ Omega X (J \ Omega)} \]

\ [\ Tag {21} \ boxed {\ int _ {-\ infty} ^ {t} X (\ Tau) d \ Tau \ overset {\ displaystyle {\ mathcal {f }}{\ leftrightarrow} \ frac {1} {J \ Omega} X (J \ Omega) + \ pi x (0) \ delta (\ Omega)} \]

3.5. Time and Frequency Scaling

If

\ [X (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (J \ Omega) \]

\ [\ Tag {22} \ boxed {x () \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow }\ frac {1 }{| A |}x (\ frac {J \ Omega} {})} \]

If order \ (A =-1 \),

\ [\ Tag {23} \ boxed {x (-T) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (-J \ Omega) }\]

3.6. Parity

3.7. passal Theorem

If

\ [X (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} X (J \ Omega) \]

Then

\ [\ Tag {24} \ boxed {\ int _ {-\ infty} ^ {+ \ infty} | x (t) | ^ 2dt = \ frac {1} {2 \ PI} \ int _ {-\ infty} ^ {+ \ infty} | x (J \ Omega) | ^ 2D \ Omega} \]

3.8. convolution

\ [\ Tag {25} \ boxed {Y (t) = h (t) * x (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} y (J \ Omega) = H (J \ Omega) x (J \ Omega )} \]

The convolution of two signals in the time domain is equal to the product of their Fourier transformation.

3.9. Multiplication

\ [\ Tag {27} \ boxed {r (t) = S (t) p (t) \ overset {{\ displaystyle {\ mathcal {f }}}{\ leftrightarrow} r (J \ Omega) = \ frac {1} {2 \ PI} [S (J \ Omega) * P (J \ Omega)]} \]

The multiplication of two signals in the time domain corresponds to Convolution in the frequency domain.

4. Fourier transform properties and basic Fourier variation list

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Continuous Time Fourier Transformation