Search results
Results from the WOW.Com Content Network
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form.
Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution () differs from cross-correlation only in that either () or () is reflected about the y-axis in convolution; thus it is a cross-correlation of () and (), or () and ().
If we start with convolution ring of positive half-line functions, the above construction is identical in behavior to the Laplace transform, and ordinary Laplace-space conversion charts can be used to map expressions involving non-function operators to ordinary functions (if they exist). Yet, as mentioned above, the algebraic approach to the ...
Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If f(t) is a real- or complex-valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. [1] [2] It can be considered a discrete-time equivalent of the Laplace transform (the s-domain or s-plane). [3]
The original proof by Titchmarsh uses complex-variable techniques, and is based on the Phragmén–Lindelöf principle, Jensen's inequality, Carleman's theorem, and Valiron's theorem. The theorem has since been proven several more times, typically using either real-variable [3] [4] [5] or complex-variable [6] [7] [8] methods.
However, the Riemann–Lebesgue lemma does not hold for arbitrary distributions. For example, the Dirac delta function distribution formally has a finite integral over the real line, but its Fourier transform is a constant and does not vanish at infinity.