All Questions
Tagged with convolution summation
58
questions
6
votes
1
answer
131
views
A nested double sum(to do with e?)
I’ve come across this nested double sum while doing an investigation but cannot seem to find a closed form for it.
$$1+\sum_{i=1}^\infty{\frac{1}{i!} \sum_{j=0}^i}{\frac{1}{j!}}$$
This is about the ...
- 143
1
vote
0
answers
138
views
What is this operation?
Let $F[n]$ and $G[n]$ be arrays of length $N$. At first, $G=F$.
After initialisation, $G$ is calculated by the relation
...
- 1,119
0
votes
1
answer
64
views
Summation with a variable in lower limit and upper limit (For convolution)
I am currently trying to solve a convolution problem analytically
For the discrete-time input signal x(n) and the step response h(n)
$$
x(n) = \left.
\begin{cases}
1, & \text{for } 0 \leq ...
5
votes
3
answers
201
views
Alternating sign Vandermonde convolution like quantity
I am interested in proving that the numbers $A_{n,m}$ are non-negative:
$$A_{n,m}:= \sum_{n_1=0}^{2n-1} \sum_{m_1=0}^{2m-1} (-1)^{m_1+n_1} \binom{2n-1-n_1}{2m-1-m_1} \binom{n_1}{m_1}$$
where $m,n$ are ...
- 53
1
vote
2
answers
73
views
What is the closed form sum of the convolution of non-zero squares?
I am trying to do the runtime analysis for a coding problem, and I've found that the total number of operations is equal to:
\begin{equation}
\sum_{i=1}^{n} i^2 (n-i)^2
\end{equation}
Where n is the ...
0
votes
0
answers
48
views
Multiplication of a power series and a finite-order polynomial [duplicate]
I am trying to find a general expression for the coefficients of the power series that results from the multiplication of a polynomial and a power series. I have looked at this post Convolution and ...
- 317
1
vote
0
answers
59
views
Minimum of a finite Fourier sum
I would like to find the minimum $$m_N = \min_{0\leq x\leq 2\pi} \{ f_N(x) \} \quad \hbox{ with } \quad f_N(x)=\frac{N}{2}+\sum_{k=1}^{N-1} (N-k) \cos(k x)$$ for all integers $N\geq 2$. It looks like ...
- 11
2
votes
1
answer
276
views
How to prove that the convolution product is commutative?
$ x\in \mathbb Z $
$$(f*g)(x)= \sum_{k=0}^{N-1} f(k) g(x-k).$$
How can I prove the commutativity of the convolution product with this expression and without integrals?
I tried with the substitution $j=...
- 23
1
vote
1
answer
34
views
Understanding the mathematical relation in the convolution sums
I am looking at an example on convolution sums. In the example it states the following:
$$ y[n] = \sum_{k=-\infty }^{n }2^{k} = \sum_{m=0}^{\infty } \left ( \frac{1}{2} \right )^{m-n}$$
The sum for $2^...
- 217
1
vote
0
answers
64
views
$d\lambda$ in the convolution integral vs its counterpart the convolution sum
So for a bit of background, in my courses I have been presented to convolution integral and then convolution sum with the following formulas :
Continuous case :
$$x(t)*h(t) = \int_{-\infty}^{\infty} x(...
- 111
3
votes
1
answer
195
views
Behavior of self convolution of 1/n
I wish to find a closed form, or a good upper bound for $\sum_{i=1}^{i=n-1} \frac{1}{i \times (n-i)}$.
I can specify a lower bound of $(n-1)/n^2$, which looks like $1/n$ because we have $n-1$ terms ...
- 698
1
vote
1
answer
186
views
How to prove that convolution of sequences is associative?
Let {$a_n$} and {$b_n$} be finite real sequences with $n\ge0$. Convolution ($\ast$) of two sequences defined as
$$
\{a_n\}\ast\{b_n\}=\{\sum_{i=0}^{n} a_ib_{n-i}\}.
$$
The convolution of three ...
- 13
2
votes
2
answers
141
views
Finite convolution sum of power function
I am interested in understanding the behavior of the convolution sum $$\sum_{k = 1}^{n} k^r (n - k)^s$$ for integer $n$, $r$, and $s$. I imagine that there must be some larger theory surrounding sums ...
- 93
0
votes
1
answer
833
views
The mean of exponential distributions
I would like to ask the following question. Let's say we have two independent variables of x and y, and both are exponentially distributed with mean $\mu_1, \mu_2$. And let $S=x+y$. May I know whether ...
- 633
0
votes
1
answer
35
views
Periodic functions summation order
It can be shown, if I am not mistaken, that for $g(n) \geq n,\; n \in \mathbb{N}$ holds:
\begin{equation}
F(n,g(n)) = \sum_{d=1}^{n}\sum_{k=1}^{g(n)} f(k,d) \sim \sum_{d=1}^{n} \frac{g(n)}{d}\sum_{...
- 265