The Fourier transform simply states that any non-periodic signal which has finite area under the curve can be represented into integrals of the sines and cosines after being multiplied by a certain weight. The Fourier transform is given by
The inverse is given as
Conditions for Fourier Transform
We can find Fourier transform of a function if it satisfies Dirichlet’s condition. For the existence of its Fourier Transform. The conditions are
- The function must have finite maxima and minima.
- In a given interval of time the function should have finite discontinuities.
- The function must be integrable at a given interval of time.
Discrete Time Fourier Transform
The sequence in discrete time can be represented by x[n] and the sequence terms are of complex exponential terms as shown below.
The DTFT sequence is as shown
The Inverse DTFT is as shown
Condition for Convergence
The term x[n] in the below series is always summable irrespective of the fact whether the series converges or not.
Then the sequence will have finite energy if it is absolutely summable and most importantly, finite energy sequence is not always absolutely summable.
Properties
If x(t)↔ X(⍵) and the output y(t)↔Y(⍵). Then the properties of Fourier Transform are
Linearity property
Then this property follows the superposition theorem. Moreover the linearity property shows that
Time Shifting Property
The time-shifting property means that a shift in time corresponds to a phase rotation in the frequency domain.
Frequency Shifting Property
Secondly, This property shows that multiplication in frequency domain results in frequency shifting
Time Reversal Property
Lastly the time reversal property is as shown
Time Scaling Property
This property shows that if a function is expanded in time by a quantity a , the F.T is compressed in frequency by the same amount.
Hence, we can write the property is as shown in equation.
Differentiation Property
Integration Property
Convolution Property
Application
Their main application is in image compression, filtering and image analysis.
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