# Advanced Signals and Systems - Linear and Cyclic Convolution

### 17. Linear convolution of sequences.

Find the convolution sum $$v(n) = v_1(n) \ast v_2(n)$$ of the following sequences $$v_1(n)$$ and $$v_2(n)$$

\begin{align} v_1(n) =& \rho _1^n \cdot \gamma_{-1}(n) \nonumber \\ v_2(n) =& \rho _2^n \cdot \gamma_{-1}(n)\nonumber \end{align}

where $$0 < \rho_1$$ and $$\rho_2 < 1$$.

## Amount and difficulty

• Working time: approx. xx minutes
• Difficulty: xx ## Solution

\begin{align*} v(n) &= \sum \limits_{k=-\infty}^{\infty} v_1(k) \cdot v_2(n-k) \\ &= \sum \limits_{k=-\infty}^{\infty} \rho_1^k \cdot \gamma_{-1}(k) \cdot \rho_2^{n-k} \cdot \gamma_{-1}(n-k) \\ &= \cdots\\ &= \begin{cases} \frac{\rho_1^{n+1}-\rho_2^{n+1}}{\rho_1-\rho_2}\cdot \gamma_{-1}(n) &, \rho_1\neq \rho_2\\ \rho_2^n\cdot(n+1)\cdot \gamma_{-1}(n) &, \rho_1 = \rho_2 \end{cases}\end{align*}

### 18. Linear and cyclic convolution.

Given two sequences $$v_1(n)$$ and $$v_2(n)$$ of length $$M=5$$:

\begin{align} v_1(n) =& [5,4,3,2,1] \nonumber \\ v_2(n) =& [1,2,3,4,5]\nonumber \end{align}

Determine the linear convolution $$v_3(n)$$ and the cyclic convolution $$v_4(n)$$ of the sequences. Give a method to calculate the linear convolution.

## Amount and difficulty

• Working time: approx. xx minutes
• Difficulty: xx

Linear convolution Cyclic convolution Calculating the linear convolution by cyclic convolution: ## Solution

• Linear convolution

\begin{align*} v_3(n) &= v_1(n) * v_2(n) = \sum \limits_{k=-\infty}^{\infty} v_1(k) \cdot v_2(n-k)\\ &\cdots\\ v_3(n) &= [5, 14, 26, 40, 55, 40, 26, 14, 5] \ \ \ \text{ for } \ \ 0\leq n \leq M-1 \end{align*}

• Cyclic convolution

\begin{align*} v_4(n) &= \sum \limits_{k=-\infty}^{\infty} v_1(k) \cdot v_2(\text{mod}(n-k,M))\\ &\cdots\\ v_4(n) &= [45, 40, 40, 45, 55] \end{align*}

• Modified cyclic convolution to get correct result

The cyclic convolution of $$v'_1(n)$$ and $$v'_2(n)$$ is calculated, where $$v'_1(n)$$ and $$v'_2(n)$$ denote the sequences $$v_1(n)$$ and $$v_2(n)$$ padded with $$M-M_2$$ and $$M-M_1$$ zeros, where $$M$$ is the length of the resulting sequence $$v_5(n)$$ and $$M=M_1+M_2+1$$.

\begin{align*} v_5(n) &= \sum \limits_{k=-\infty}^{\infty} v'_1(k) \cdot v'_2(\text{mod}(n-k,M))\\ &\cdots\\ v_5(n) &= [5, 14, 26, 40, 55, 40, 26, 14, 5] \ \ \ \text{ for } \ \ 0\leq n \leq M-1 \end{align*}