Signals Through Linear Shift-Invariant Systems

Definitions  A system (operator) H is called linear if for every two signals and constants a,b:  A system (operator) H is called shift-invariant (or time-invariant) if for each signal :  Operators can be both linear and SI, or neither. It is also possible for an operator to be SI but not linear, or linear but not SI.

The Impulse Response of an LSI System  The impulse response of an LSI system H, is the result of H operating on a delta function. It is often marked

Proporties of LSI Systems  Every LSI operation can be expressed as a convolution between the input signal and the system's impulse response. Mathematically: for each LSI system H and for each input signal :  Proof: Let H be an LSI system and let be its input signal.

Proporties of LSI Systems  Let H1,H2 be two LSI systems, then:  Proof:

LSI Operators On Orthonormal Families  Let be an orthonormal family.  Let. We want to check if are orthonormal.  If by some miracle this family satisfies such that then the inner product would be: we say that the functions are eigenfuctions of H and are their eigenvalues.

The Majestic Family  Let H be an LSI system, then for each, where Proof:

The Majestic Family  As a result, when the signal (the representation of the signal in the Fourier basis) goes through an LSI system, the result is also a linear combination of with new coefficients :  Another important conclusion is the following theorem:  Proof:

Example of Medical Usage of LSI Ops. Laplacian filter of bone scan (a) Sharpened version of bone scan achieved by subtracting (a) and (b) Sobel filter of bone scan (a) (a) (b) (c) (d) Meet Bob!

Example of Medical Usage Sharpened image which is sum of (a) and (f) Result of applying a power-law trans. to (g) (e) (f) (g) (h) The product of (c) and (e) which will be used as a mask Image (d) smoothed with a 5*5 averaging filter

BEFORE: AFTER: