# BER calculation in DS-UWB system

I need to calculate BER (bit error rate) for a system using spreading spectrum DS over ultra wideband (UWB). Modulation is bpsk. The spreading sequence lengths are not equal for all users. Receiver is matched filter. I know the equations of DS-UWB transmission signal and template signal which I should use at receiver to extract the user's data. I know that I should use convolution between the received signal (transmitted + AWGN noise) and template signal. My problem is, how to calculate the result of that convolution? is it integration or multiplication? thanks in advance

• It's not so clear what you're asking. Convolution is...convolution. It involves multiplicaton and integration. – Deve Jun 29 '13 at 10:45
• @Deve thanks for response. Can you tell me some reference where I can find details how mathematically the convolution output is calculated for ds-uwb? – Mohammed Jul 1 '13 at 9:49
• I'm afraid I can't. What makes the convolution in DS-UWB different from a conventional convolution? – Deve Jul 1 '13 at 11:50
• @Deve Actually this is my problem, I don't know how to calculate convolution between two sequences with different lengths. – Mohammed Jul 1 '13 at 12:46

## 1 Answer

Assume two discrete-time sequences: $x$ (length $L_\mathrm{x}$) and $y$ (length $L_\mathrm{y}$). Furthermore, $L_\mathrm{x} \geq L_\mathrm{y}$. To calculate the discrete-time (linear) convolution of $x$ and $y$, first pad $y$ with $L_\mathrm{x} - L_\mathrm{y}$ zeros, i.e. $$y'(n) = \begin{cases} y & 0 \leq n < L_\mathrm{y}\\ 0 & L_\mathrm{y} < n \leq L_\mathrm{x} - 1 \end{cases}$$ Then calculate the convolution of $y'$ and $x$ as described here, for example.

• The zero-padding is not required. It will just increase the amount of multiplications and since most of them will have one factor of zero, nothing is gained. – jan Aug 1 '13 at 9:55