New answers tagged linear-systems
0
I think the answer to your question is "yes" and this is why:
A Linear Time-Variant (or possibly time-variant) system is fully described (from the POV of output $y(t)$ in terms of input $x(t)$) by this convolution integral:
$$ y(t) = \int\limits_{-\infty}^{\infty} h(t,u) x(u) \, \mathrm{d}u $$
$h(t,u)$ is the impulse response function, evaluated at time $...
1
LTI means Linear Time-Invariant systems. The system has to satisfy two conditions.
(1) Linear and (2) Time-Invariant.
(1) Linear means, if the response of the system due to load Px and Py is Rx and Ry respectively, then for the load (Px+Py), the response of the system will be (Rx+Ry).
(2) Time-Invariant means, the parameters of the system does not vary with ...
1
To answer your second question: To determine the cross correlation with discrete tones you can get the frequency response of your correlation by treating the sequence as the time reversed coefficients of an FIR filter (since the FIR filter performs convolution of your signal with the coefficients, and correlation is convolution with one of the sequences time ...
2
I assume that this is a discrete-time problem where the maximum-length sequence is a pseudorandom sequence $x[k]$ of $\pm 1$ values and the noise is a sequence $n[k]$ of independent identically distributed (iid) zero-mean random variables with variance $\sigma^2$. Then, $\sum_{k=0}^{N-1} x[k]n[k]$ is also a sum of $N$ iid random variables and its variance ...
2
Let's go through a few ways to solve this:
Fourier transform: an ideal integrator is an LTI system, so its response to a sinusoidal input signal is a sinusoid with the amplitude and phase changed according to the frequency response evaluated at the input frequency (if it exists). For the ideal integrator we have
$$H(\omega)=\pi\delta(\omega)+\frac{1}{j\...
2
With feedback systems such as the one given it's often easy to define an additional signal at the output of the adder. This gives the following equations:
$$U(f)= X(f)-H_2(f)Y(f)\tag{1}$$
and
$$Y(f)=U(f)H_1(f)\tag{2}$$
Now you can solve Eqs $(1)$ and $(2)$ to get the frequency response $H(f)=Y(f)/X(f)$.
2
Your math looks correct, thanks for including what you have done. Also for such a block diagram of a linear system, you can rearrange each of the three blocks in any order at the points where the nodes come together (can't break loops) - For example, you can move the derivative to the end without changing the overall result. This may make it even more ...
0
You are right.
In the solution, the second line which performs the long division
$$ H(z) = -4 + \frac{ 5 + \frac{7}{2} z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} } $$
is wrong and should be corrected as:
$$ H(z) = -4 + \frac{ 5 - 3 z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} } .$$
However, the partial fraction expansion at the following ...
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