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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 ().
For example, when = and =, Eq.3 equals , whereas direct evaluation of Eq.1 would require up to complex multiplications per output sample, the worst case being when both and are complex-valued. Also note that for any given M , {\displaystyle M,} Eq.3 has a minimum with respect to N . {\displaystyle N.} Figure 2 is a graph of the values of N ...
Convolution is a linear operation. It then follows that the multidimensional convolution of separable signals can be expressed as the product of many one-dimensional convolutions. For example, consider the case where x and h are both separable functions.
Let G be a group and k a field.The group Hopf algebra of G over k, denoted kG (or k[G]), is as a set (and a vector space) the free vector space on G over k.As an algebra, its product is defined by linear extension of the group composition in G, with multiplicative unit the identity in G; this product is also known as convolution.
Also, the output of a linear system can contain harmonics (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by () = (+ ()) (). It is linear because it satisfies the superposition principle.
Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix. Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity. For symmetric Toeplitz matrices, there is the decomposition
For example, when = and =, Eq.3 equals , whereas direct evaluation of Eq.1 would require up to complex multiplications per output sample, the worst case being when both and are complex-valued. Also note that for any given M , {\displaystyle M,} Eq.3 has a minimum with respect to N . {\displaystyle N.} Figure 2 is a graph of the values of N ...
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).