1
vote
0answers
41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
2
votes
1answer
62 views

Moving average as ODE

Is it possible to represent or approximate the moving average $m(t) = \frac{1}{w}\int_{t-w}^t x(\tau) d\tau$ of a function $x(t)$ as a set of ordinary differential equations $\dot{y} = \ldots$? I am ...
2
votes
3answers
182 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
2
votes
1answer
91 views

Multiplication with the derivative of the dirac delta

I have a function $x(t)$ that I'm multiplying with $\frac{d}{dt}\delta(t-kT)$ I know the property that $\frac{d}{dt}\delta(t-kT) = -\frac{\delta(t-kT)}{t-kT}$, and if I use that: ...
1
vote
1answer
32 views

Should mean be subtracted before conducting singular spectrum analysis (SSA)?

I have read that for the multivariate form you need to subtract the mean and divide by the standard deviation. Is this necessary before performing basic SSA on one signal? Thanks
4
votes
0answers
98 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
1
vote
1answer
53 views

Fourier analysis of an exponential function review

I am working through and reviewing some of the examples presented on Fourier analysis from a Modern Digital and Analog Communication Systems book. In one of the examples, the author goes through the ...
1
vote
0answers
32 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
0
votes
1answer
30 views

Elimination of complex variable in integral

I have the equation: $$\frac{1}{\tau}\intop_{0}^{\tau}A\sin\left(\Omega t\right)\cdot A\sin\left(\Omega\left(t-\lambda\right)\right)\mathrm{d}t$$ for which the attempted solution is to convert the ...
17
votes
3answers
579 views

Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using ...
2
votes
3answers
983 views

Proofs of dirac delta property

How would I formally prove this property of dirac delta? $$\int \delta(a-x) \delta(x-b) \,dx = \delta(a-b) $$ I attempted to use the definition of a dirac delta $$\int ...
2
votes
2answers
239 views

I need to calculate the period, I need help to verify my answer

I need to determine if $x(t) = 9\cos(2t) + 4\sin(\pi t)$ is periodic. If it is periodic I need to find the period. this what I have done \begin{align*} T_0 &= 2\pi/w\\ T_1 &= 2\pi/2 = \pi \\ ...
0
votes
0answers
328 views

Fourier Series Coefficients for Signals

The question is: We specify the fourier series coefficients of a continuous-time signal that is periodic with period 4. Determine the signal x(t). $a_k=\begin{cases} 0, & k=0\\ ...
0
votes
1answer
116 views

Fourier Series with Signals

So the question is: Determine the fourier series representations for the following signal: Here the formula for the fourier series $$C_k=\frac{1}{T}\int_T \! x(t)e^\frac{-j2\pi kt}{T} \, \mathrm{d} ...
7
votes
2answers
11k views

Integration of sawtooth, square and triangle wave functions

Context After a discussion about how to plot the results of a frequency modulation between two signals on Stack Overflow, I understood that I need to find the time-integral of the following wave ...
3
votes
1answer
916 views

Derivative of a random variable w.r.t. a deterministic variable

I'm reading about time series and I thought of this procedure: can you differentiate a function containing a random variable. For example: $f(t) = a t + b + \epsilon$ where $\epsilon \sim N(0,1)$. ...
2
votes
2answers
229 views

Can it be proven that such functions don't exist?

We are given $x_1,x_2 \in \mathbb{R}$ and we want to find two functions $v_1(t),v_2(t)$ such that: $$x_1x_2 = \int_{-\infty}^{\infty} v_1(t)-v_2(t) dt$$ A very interesting restriction that we have ...
1
vote
3answers
2k views

How to sketch a sinc function by hand?

I have to do this for an upcoming exam, but cannot find anywhere (in the textbook or online) how to do this. I only really need to know a couple points to plot it... when x = 0, and then the earliest ...
1
vote
2answers
86 views

Preserving the extrema of one function after applying another

Suppose we have some function $f(x)$ with local extrema at $x_1, x_2, \dots$, and a second function $g(x)$ which is continuous, strictly increasing and non-zero everywhere over the range of the $x_i$. ...