Question

If the Eigen function of an LTI system is x(n)= Ae^jnπ and the impulse response of the system is h(n)=(1/2)^nu(n), then what is the Eigen value of the system?

(a) 3/2

(b) -3/2

(c) -2/3

(d) 2/3

I got this question in a national level competition.

I want to ask this question from Frequency Domain Characteristics of LTI System topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right choice is (d) 2/3

The best explanation: First we evaluate the Fourier transform of the impulse response of the system h(n)

H(ω)=\(\sum_{n=-∞}^∞ h(n) e^{-jωn} = \frac{1}{1-\frac{1}{2} e^{-jω}}\)

At ω=π, the above equation yields,

H(π)=\(\frac{1}{1+\frac{1}{2}}\)=2/3

If the input signal is a complex exponential signal, then the input is known as Eigen function and H(ω) is called the Eigen value of the system. So, the Eigen value of the system mentioned above is 2/3.