Question

For the system \( y (t) = ∑_{k=-∞}^n x(k)\), which of the following holds true?

(a) Invertible

(b) Non-Invertible

(c) Invertible as well as Non-Invertible in its respective domains

(d) Cannot be determined

I had been asked this question in a national level competition.

Question is taken from Properties of Systems topic in chapter Signals and Systems Basics of Signals and Systems

Answer 1

The correct option is (a) Invertible

The explanation is: \( y (t) = ∑_{k=-∞}^n x(k)\)

Or, y (n) = x (n) + x (n-1) + x (n-2) + …….

Or, y (n+1) = x (n+1) + x (n) + (terms after this)

Or, y (n+1) = x (n+1) + y (n)

Or, y (n+2) = x (n+2) + y (n+1)

Or, y (n) = x (n) + y (n-1)

This is an alternate form of given system equation.

∴ x (n) = y (n) – y (n-1)

Taking z-transform on both sides,

X (z) = Y (z) = (1-z-1)

\(H (z) = \frac{Y(z)}{X(z)} = \frac{1}{1-z-1}\)

Or, \(\frac{w(z)}{y(z)} \)= H^-1(z) = (1-z-1)

∴ w (n) = y (n) – y (n-1)

Hence, the system is invertible.