0.10 Pulse shaping and receive filtering  (Page 12/13)

Let the pulse shape be a $T$ -wide Hamming blip. Use the code in matchfilt.m to find the ratio of the power in the downsampled $v$ to that in the downsampled $w$ when:

1. the receive filter is a SRRC with beta= 0, 0.1, 0.5,
2. the receive filter is a rectangular pulse, and
3. the receive filter is a $3T$ -wide Hamming pulse.

When is the ratio largest?

Let the pulse shape be a SRRC with beta=0.25 . Use the code in matchfilt.m to find the ratio of the power in the downsampled $v$ to that in the downsampled $w$ when

1. the receive filter is a SRRC with beta= 0, 0.1, 0.25, 0.5,
2. the receive filter is a rectangular pulse, and
3. the receive filter is a $T$ -wide Hamming pulse.

When is the ratio largest?

Let the symbol alphabet be 4-PAM.

1. Repeat [link] .
2. Repeat [link] .

Create a noise sequence that is uniformly distributed (using rand ) with zero mean.

1. Repeat [link] .
2. Repeat [link] .

Consider the baseband communication system in [link] . The symbol period $T$ is an integer multiple of the sample period $T_s$ , i.e.  $T=N{T}_s$ . The message sequence is nonzero only each $N$ th $k$ , i.e.  $k=N{T}_s+\delta T_s$ where the integer $\delta$ is within $0\le \delta <N$ and $\delta T_s$ is the “on-sample” baud timing offset at the transmitter.

1. Suppose that $f\left[k\right]=\{0.2,0,0,1.0,1.0,0,0,-0.2\}$ for $k=0,1,2,3,4,5,6,7$ and zero otherwise. Determine the causal $g\left[k\right]$ that is the matched filter to $f\left[k\right]$ . Arrange $g\left[k\right]$ so that the final nonzero element is as small as possible.
2. For the $g\left[k\right]$ specified in part (a), determine the smallest possible $N$ so the path from $m\left[k\right]$ to $y\left[k\right]$ forms an impulse response that would qualify as a Nyquist pulse for some well-chosenbaud-timing offset $\sigma$ when $y$ is downsampled by $N$ .
3. For the $g\left[k\right]$ chosen in part (a) and the $N$ chosen in part (b), determine the downsampler offset $\sigma$ for $s\left[i\right]$ to be the nonzero entries of $m\left[k\right]$ when $w\left[k\right]=0$ for all $k$ .