linear_systems

# We claim that e j 2 π fx is an eigenfunction of

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We claim that e j 2 π fx is an eigenfunction of continuous LTI systems. If v ( x )= e j 2 π fx is an eigenfunction, then L braceleftBig e j 2 π fx bracerightBig = λ e j 2 π fx . We can check that indeed e j 2 π fx is an eigenfunction of the LTI system, by substituting this function into the convolution integral. L braceleftBig e j 2 π fx bracerightBig = integraldisplay h ( z ) e j 2 π f ( x z ) dz = e j 2 π fx bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright eigenfunction integraldisplay h ( z ) e j 2 π fz dz bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright eigenvalue H ( f ) (at frequency f ) H ( f ) is the Fourier transform of the impulse response of the system. Complex exponentials are the eigenfunctions of LTI systems, and their eigenvalues are the Fourier transform of the impulse response of the system. 4 Transforms We have seen that complex exponentials are eigenvectors and eigenfunctions respectively of discrete and continuous LTI systems. Therefore, a LTI system can only scale a complex exponential, where the scaling factor is the (frequency dependent) eigenvalue. (Note that the eigenvalue could be complex which would indicate a phase shift at that frequency.) In general, signals are not going to be complex exponentials: for instance, they are likely to be real-valued. What we do to analyze LTI systems is to write the signal as a linear combination of complex exponentials . How do we write a signal as a linear combination of complex exponentials? The Fourier transform expresses a signal as a linear combination of complex exponentials. The basis functions are the natural basis of the LTI system.

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EE 505 B, Autumn, 2011 Linear Systems 11 v ( x )= integraldisplay V ( f ) e j 2 π fx df bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright linear combination of complex exponentials where V ( f ) = integraldisplay v ( x ) e j 2 π fx dx The output of a LTI system is then just the sum/integral of the scaled complex exponen- tials, because complex exponentials are the eigenvectors/eigenfunctions of the system. 4.1 Fourier transforms The following table summarizes the definitions for various Fourier transforms. The columns of the table show continuous time versus discrete time transforms, and the rows of the table show aperiodic in time signals versus periodic in time signals. Note that we have used the notation for these transforms that you will see in courses such as EE 518. This notation differs slightly from what we have used previously, however, you should able to adapt to different notations, because there is no universal standard. Continuous time Discrete time (Continuous-time) Fourier transform Discrete-time Fourier Transform Time aperiodic x ( t ) = 1 2 π integraldisplay X ( j Ω ) e j Ω t d Ω X ( j Ω ) = integraldisplay x ( t ) e j Ω t dt x [ l ] = 1 2 π integraldisplay π π X ( e j ω ) e j ω l d ω X ( e j ω )= + l = x [ l ] e j ω l (Continuous-time) Fourier Series Discrete(-time) Fourier Series Time periodic x ( t )= + k = X [ l ] e j 2 π T kt X [ k ] = 1 T integraldisplay T 2 T 2 x ( t ) e j 2 π T kt dt x [ l ]= N 1 k = 0 X [ k ] e j 2 π N kl X [ k ] = 1 n N 1 l = 0 x [ l ] e j 2 π N kl The discrete Fourier transform (DFT) is not listed in the table, but the equations are the same as the discrete(-time) Fourier series. You’ll see the DFT in EE 518.
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