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### 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 ...
• 28.3k
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### 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 ...
• 45.4k
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### 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); ...
• 1,807
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### 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 ...
• 90.4k
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### 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 ...
• 2,584

### 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 ...
• 90.4k

### 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 ...
• 2,546
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. ...
• 90.4k

### 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 ...
• 24.6k
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### 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 $t_0>0$. ...
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### 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 ...
• 31.9k
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### 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, ...
• 90.4k
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### 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 ...
• 90.4k
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### 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 ...
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### 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......
• 28.3k

### Negative group delay and envelope advance

Here is a actual example with negative group delay that will provide further insight: Below is a plot of the output and input of a pulse through a realizable filter that has negative group delay: It ...
• 52.1k
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### 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 ...
• 2,611

### 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.
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• 90.4k
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### 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 ...
• 31.9k
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### 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 ...
• 20.4k
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### 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 ...
• 28.3k
Accepted

### How to find the difference equation directly from Direct Form II signal flow graph

Darkling, these things are quite clearly explained in standard signals & systems textbooks. But I assume you have little time left to read (as most undergraduate courseware are full of homeworks, ...
• 28.3k
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_{...