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Course by: Mark A. Davenport. E-mail the author

# Eigenbases and LSI Systems

Module by: Mark A. Davenport. E-mail the author

Why is an eigenbasis so useful? It allows us to greatly simplify the computation of the output for a given input. For example, suppose that XX is a vector space and that L:XXL:XX is a linear operator with eigenvectors {vk}kΓ{vk}kΓ. If {vk}kΓ{vk}kΓ form a basis for XX, then for any xXxX we can write x=kΓckvkx=kΓckvk. In this case we have that

y = L x = L k Γ c k v k = k Γ c k L ( v k ) = k Γ c k λ k v k y = L x = L k Γ c k v k = k Γ c k L ( v k ) = k Γ c k λ k v k
(1)

In the case of a DT, LSI system HH, we have that 12πe-jωn12πe-jωn is an eigenvector of HH and for any x[n]x[n] we can write

x [ n ] = - π π X ( e j ω ) e - j ω n 2 π d ω . x [ n ] = - π π X ( e j ω ) e - j ω n 2 π d ω .
(2)

From the same line of reasoning as above, we have that

y [ n ] = H ( x [ n ] ) = - π π X ( e j ω ) H e - j ω n 2 π d ω = - π π X ( e j ω ) H ( e j ω ) · e - j ω n 2 π d ω = - π π Y ( e j ω ) · e - j ω n 2 π d ω y [ n ] = H ( x [ n ] ) = - π π X ( e j ω ) H e - j ω n 2 π d ω = - π π X ( e j ω ) H ( e j ω ) · e - j ω n 2 π d ω = - π π Y ( e j ω ) · e - j ω n 2 π d ω
(3)

Whenever we have an eigenbasis, we can represent our operator as simply a diagonal operator when the input and output vectors are represented in the eigenbasis. The fact that convolution in time is equivalent to multiplication in the Fourier domain is just one instance of this phenomenon. Moreover, while we have been focusing primarily on the DTFT, it should now be clear that each Fourier representation forms an eigenbasis for a specific class of operators, each of which defines a particular kind of convolution.

• DTFT: discrete-time convolution (infinite)
• CTFT: continuous-time convolution (infinite)
(f*g)(t)=-f(τ)g(t-τ)dτ(f*g)(t)=-f(τ)g(t-τ)dτ
(4)
• DFT: discrete-time circular convolution
(xy)[n]=k=0N-1x[k]yN[n-k](xy)[n]=k=0N-1x[k]yN[n-k]
(5)
• CTFS: continuous-time circular convolution
(fg)(t)=0τf(t)gT(b-τ)dτ(fg)(t)=0τf(t)gT(b-τ)dτ
(6)

This is the main reason why we have to care about circular convolution. It is something that one would almost never want to do – but if you multiply two DFTs together you are doing it implicitly, so be careful and remember what it is doing.

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