Question

Are downsampling and upsampling Linear Time Invariant (LTI) systems in Digital Signal Pocessing

Answer #1

up sampling and down sampling is basically depends on time and frequency domain.

the basic methodology behind the up sampling is to
decrease the time between two instances of time ...**shown in
the picture**.

say one signal i am giving f(x) and it will be up sampled 2 times and then it will be f(y). so the relation should be like

f(y) = f(x) * 2

so like it can be up sampled N times ; so it will look like a linearly increased system.But the time is invariant as it comes under a discrete domain. as we are putting zero's to decrease time domain as frequency will increase.

same case for down sampling ; it is also a discrete signal . because to make the frequency lesser it skips instances of time one after another.

so it is also L T I system.

2. Consider the following impulse responses h[n] of linear
time-invariant (LTI) systems. In each case,
(i) provide the transfer function H(z) (ZT of h[n]) and its
ROCh,
(ii) sketch the ROCh in the z-plane,
(iii) mark the pole and zero locations of H(z) (on the same plot
in the z-plane), and (iv) discuss whether or not the LTI system is
stable.
(a) h1[n] = (0.4)^n u[n] + (2 - 3j)^n u(n -2)
(b) h2[n] = (0.2)^(n+2) u[n] + (2 -...

1. Give three linear systems and three non-linear system, and
specify whether one is time variant or time invariant.

A discrete time system can be
i. Linear or non-linear
ii. Time invariant or Time Variant
iii. Causal or noncausal
iv. Stable or unstable
v. Static Vs Dynamic
Examine the following systems with respect to every property
mentioned above and give a brief
explanation.
a. y[n] = x[n]δ[n − 1]
b. y[n] = x[n] + nu[n + 1]
c. y(n) = x(2. n)
d. y(n) = 3. x(n)

Determine if the system ?[?]=sin(0.2?∙?[?])is (1) linear, (2)
time-invariant.

Which of the following systems are dynamical? Stable? Causal?
Time-invariant? Linear?
1. y(t) = 5x(t)
2. y(t) = x(t)^4
3. y(t) = x(t−1)^2
4. dy/dt = x(t)
5. dy/dt = y(t) + x(t)
6. dy/dt = −y(t) + x(t)^2
7. dy/dt = −y(t) + x(t + 4)
8. dy/dt = tx(t)

Analytical Exercise AE8.1.
Show that observability is not invariant with respect to state
feedback.
Textbook: Linear State Space Control Systems, 2007, ISBN:
978-0-471-73555-7

CHAPTER 13: DISCRETE-TIME SIGNAL (TEXTBOOK SIGNALS AND SYSTEM BY
MAHMOOD NAHVI
11. In an LTI system, x(n) is the input and h(n) is the
unit-sample response. Find and sketch the
output y(n) for the following cases:
i) x(n) = 0.3nu(n) and h(n) = 0.4nu(n)
ii) x(n) = 0.5nu(n) and h(n) = 0.6nu(n)
iii) x(n) = 0.5|n|u(n) and h(n) = 0.6nu(n)

what are the 3 transforms in digital signal processing
give their time & frequency variables, formula,
inverse formula and application.

Explain the difference between an analog and a digital signal.
Give 2 examples of real-world digital signals.

The signal x[n] is the input of an LTI system with impulse
function of h[n]. x[n] = (0.4)^n u[n] and h[n] = (0.2)^n u[n].
(a) What is the DTFT of the output of the LTI system?
(b) What are the Energy density spectrums of the input and
output signals?
(c) What would be the inverse DTFT: X(w) =
1/(1-0.25e^-j(w-2))
(d) How would part (c) differ for the DTFT: X(w) =
1/(1-0.25e^-j(w-2)) + 1/(1-0.25e^-j(w+2))

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