Question

1. (15 points) Review Carefully read each statement below and identify it as True or False and, for each answer, explain your reasoning briefly in 1-2 sentences. 1. A 100KHz tone (i.e., sinusoid) is input to an LTI system. The frequency spectrum of the output signal may include components at integer multiples of 100KHz. True or False? Why? 2. The phase of the Fourier transform of a real-valued signal with odd symmetry will always be +90 deg. True or False? Why? 3. The input-output of an LTI system is fully characterized once we know its impulse response. True or False? Why? 4. The impulse response sequence of a discrete-time accumulator is absolutely summable. True or False? Why? 5. The output of a linear time-varying system to a periodic input signal will always be periodic with the same fundamental period as the input. True or False? Why? 6. The signal a sin(wit) + a2 sin( wt) is always periodic for any values of frequencies wi and we True or False? Why? 7. A periodic signal with even symmetry (i.e., z(t) <(-)) will always be even harmonic (i.e., its Fourier series coefficients ax = 0 for k odd). True or False? Why? 8. An LTI system must be BIBO stable for us to be able to obtain and characterize its frequency response. True or False? Why? 9. Any signal that has finite energy will be absolutely integrable too. True or False? Why? 10. The magnitude frequency spectrum of any real-valued signal will always have even symmetry with respect to frequency. True or False? Why?

Answer 1

(1).

False.

Input Output relation:

x(t) => x(nt)

where n is integer

If we add delay in the input

x(t-k) => Y=x(n(t-k)) = x(nt-nk)

If we add delay at the output

Y'=x(nt-k)

So, Y != Y'

Time variance violates.

So, an LTI system can not provide output with the frequency of integer multiple of the input frequency.

(2).

True.

The frequency spectrum of an odd symmetry can be written as:

f(k) i/2 A(k)e'

_$e^{i\pi/2}$ represents the 90 degree phase difference.

(3).

True.

Response of an LTI system can be calculated as the convolution sum of input to the impulse response.

$y(t)=x(t)*h(t)$

(4).

True.

Discrete-time accumulator is a linear system. the impulse response is finite and absolutely summable.

(5).

True.

For time invariance:

if y(t) is the response to an input signal x(t), then y(t+T) will be the output to the shifted input x(t+T). If the input signal x(t) is periodic with a period of T, this suggests that x(t)=x(t+T), and, consequently, y(t)=y(t+T), i.e. the output is also periodic with a same time period as the input signal.

(6).

False.

For the signal to be periodic, the ratio $\frac{w_1}{w_2}$ must be rational.

(8).

True.

To apply frequency response the system response must be absolutely integrable, and for that the system must be BIBO stabel.

(9).

True.

Energy of a signal:

En = L ( 2 (t)D’dt

From the expression above we can clearly conclude, for the energy to be finite it must be integrable over the period - infinity to + infinity.