17
votes
Accepted
Why LTI system cannot generate new frequencies?
One of the definitive features of LTI systems is that they cannot generate any new frequencies which are not already present in their inputs.
One way to see why this is so, comes by observing the ...
- 28k
15
votes
Accepted
If the convolution of two signals is a unit impulse, what does this tell us?
It tells us that the systems are inverses of each other. The DFT of
$$h_1[n]*h_2[n]= \delta[n]$$
is
$$H_1[k] \cdot H_2[k] = 1 $$
so we get
$$H_2[k] = \frac{1}{H_1[k]}, H_1[k] = \frac{1}{H_2[k]}$$
In ...
- 42.5k
13
votes
Accepted
What is the difference between natural response and zero input response?
First it's important to realize that many authors use the terms zero-input response and natural response as synonyms. This convention is used in the corresponding wikipedia article, and for instance ...
- 88.8k
12
votes
Accepted
Why $y[n] = x[-n]$ is not time-invariant?
A time-invariant system is one that, when you shift the input signal, the output is shifted by the same amount.
A system that reverses the signal cannot be time-invariant because when you shift the ...
- 2,360
12
votes
Are all LTI systems invertible? If not, what is a good counterexample?
You need to define what you mean by "invertible". Do you mean invertible by a causal and stable system? If yes, then any system that is not minimum-phase is not invertible (because the inverse system ...
- 88.8k
12
votes
Are there any real world applications for complex-valued signals or impulse responses?
Absolutely! Conjugates are mentioned in textbooks because conjugation has no effect on real signals, but it does on complex ones. This way, formulations are more general and apply to both real and ...
- 2,326
12
votes
Accepted
What is the adjoint of a linear operator and why is it useful?
Here's the best practical information I have so far on linear operators and their adjoints. There's only one book I've come across that discusses this very practically (which I reference below); ...
- 1,652
11
votes
Accepted
How can an impulse generate an output in the past time frame?
As mentioned in SakSath's answer a system with $h[n]\neq 0$ for $n<0$ is non-causal. Such a system cannot be implemented in real-time. However, you could use such a system for offline processing. ...
- 88.8k
10
votes
Why LTI system cannot generate new frequencies?
You can make a simple algebraic argument, given the premise that you provided. If:
$$
Y(\omega) = X(\omega) H(\omega)
$$
where $X(\omega)$ is the spectrum of the input signal and $H(\omega$) is the ...
- 24.5k
10
votes
Accepted
Non-causality of fractional delays
It's not the delay itself that causes the total discrete-time system to be non-causal. In continuous time we simply have an impulse response $\delta(t-t_0)$, which is clearly causal for $t_0>0$. ...
- 88.8k
9
votes
A system that perfoms Fourier Transform operation - is it an LTI system?
The Fourier transform operator $\mathscr{F}$ is a linear one; i.e.,
$$\mathscr{F}\{x(t)\}=X(f) ~,~ \mathscr{F}\{y(t)\}=Y(f) \implies \mathscr{F}\{\alpha x(t) + \beta y(t) \} = \alpha X(f) + \beta Y(...
- 28k
8
votes
Accepted
find impulse response from step response
You can find the impulse response
Let's take the case of a discrete system.
If $s[n]$ is the unit step response of the system, we can write
$$s[n]= u[n]\ast h[n]$$
where $h[n]$ is the impulse ...
- 1,022
8
votes
Accepted
How to determine if the system is invertible
It depends on what exactly you mean by "invertible". In system theory, what is often meant is if there is a causal and stable system that can invert a given system, because otherwise there might be an ...
- 88.8k
8
votes
Is the first derivative operation on a signal a causal system?
If the derivative exists at the given point, then it doesn't matter if you look (infinitesimally) into the future or into the past, you can do both, because both will give the same result:
$$x'(t)=\...
- 88.8k
8
votes
Are all LTI systems invertible? If not, what is a good counterexample?
A necessary condition for invertibility is that any output has only one possible input (or injectivity, as proposed in comments). Since we are looking at counterexamples, we can look at when this ...
- 31.7k
7
votes
Accepted
How to obtain impulse response from the differential equation of a system?
It looks like your transfer function is correct, but there's a small mistake in your partial fraction expansion:
$$H(s)=\frac{2}{(s+4)(s+2)}=\frac{1}{s+2}-\frac{1}{s+4}\tag{1}$$
The corresponding ...
- 88.8k
7
votes
Accepted
In the context of transfer functions, what is the relationship between the terms "proper", "causal", and "realizable"?
Causality is a necessary condition for realizability. Stability (or, at least, marginal stability) is also important for a system to be useful in practice.
For linear time-invariant (LTI) systems, ...
- 88.8k
7
votes
Accepted
Why does reversing the order of these two transfer functions give me different outputs?
If the input is a unit step, then the output of the first block in system 1 is not zero, but it is a Dirac delta impulse $\delta(t)$. Intuitively, the derivative is infinite at $t=0$ because of the ...
- 88.8k
7
votes
Accepted
why exponential term neglected in equation?
The magnitude of that complex exponential is 1. Recall from complex algebra: any complex number can be expressed as $z = r e^{j \phi}$ where $|z|=r$ is its magnitude and $\arg z = \phi$ is the ...
- 4,124
7
votes
Why is the impulse response function of this system 0?
This system
$$ y(t) = t^2 x(t) $$
is not LTI and therefore does not have an impulse response of the form $h(t) = \mathcal{T}\{\delta(t)\}$.
So your statement $h(t) = t^2 \delta(t)$ is not correct......
- 28k
7
votes
Accepted
Negative group delay and envelope advance
Answer : No, any causal LTI system with frequency response $H(f)$ cannot produce the output $y(t)$ in advance. And, the answer lies in the causality of input signal $x(t)$ being applied to $h(t)$. Any ...
- 2,591
7
votes
Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?
Consider the identity system $y(t) = x(t)$. This system is LTI. If the input $x(t)$ is discontinuous, then the output $y(t)$ will be discontinuous too.
- 14.9k
7
votes
Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?
To add an even worse example to MBaz (best possible) counterexample:
The derivative $\frac{\mathrm d}{\mathrm dt}$ is an LTI system. $f(t)=|t|$ is a continuous function.
$\frac{\mathrm d}{\mathrm dt}(|...
- 29.6k
6
votes
Accepted
Initial conditions for the LTI systems described as a difference equations
The problem is that non-zero initial conditions cause a term in the output signal that does not depend on the input signal. This explains why a system with non-zero initial conditions can neither be ...
- 88.8k
6
votes
Accepted
$\mathcal Z$-transform ROC
If the ROC is outside a circle in the complex $z$-plane ($|z|>a$), then the corresponding system is causal. If it is inside a circle ($|z|<a$), the system is anti-causal. If the ROC is a ring ($...
- 88.8k
6
votes
Accepted
What Is the Transfer Function of a Moving Average (FIR Filter)?
The frequency response of the moving average is called the asinc or psinc, the aliased sinc or periodic sinc (sinc for cardinal sine), or the Dirichlet function.
Since the sum of the moving average ...
- 31.7k
6
votes
Accepted
Help in understanding the formula of Signal-to-Noise-Ratio (SNR) - Part 1
$\mathsf{SNR}$ (signal-to-noise ratio) is a generic term whose value can be defined in different ways by different people, and as long as one states clearly what is meant by $\mathsf{SNR}$ in a ...
- 20.2k
6
votes
Accepted
Why is $y(t)=x(t/2)$ a non-causal system?
Because for negative values of $t$ you have, for example, $y(-2) = x(-1)$ which depends on a future value of $x(t)$ at $t=-1$ for the current value of $y(t)$ at $t=-2$.
Note that $t=-1$ represents a ...
- 28k
6
votes
Can every type of linear filter be modelled by a convolution?
No. It's only LTI (Linear and Time-Invariant) systems that can be modeled with convolution through a unique single impulse response.
For example the systems
$$ y(t) = g(t) x(t) $$ or
$$ y[n] = \sum_{...
- 28k
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