IE$3012$

$4.($a$)$ Assume that the error amplitude, $q,$ due to threshold of coding in a singleintegration Delta Modulator $($DM$),$ has a uniform distribution and lies in the range $(-,+)$ and a uniform power spectral density $($PSD$)$ concentrated over $(0,f_s)$ where $f_s$ is the sampling frequency. The DM is used to accept a baseband analogue signal input with maximum bandwidth of B Hz $.$ If the baseband filter of the Delta demodulator has the transfer function as shown in Figure $1,$ show that the average quantization noise power can be expressed as $\frac{?}{\text{b}\text{a}\text{r}}(q^2)=\frac{{}^2}{6f_s}(2B+).$

$(10$ Marks$)$

Figure $1$

$($b$)$ If the input baseband analogue signal of the DM in part $($a$)$ has a bandwidth of $B=4$ kHz and an average power of $14$ W $,$ and $=1\mathrm{.}05,f_s=120kHz$ and the Delta demodulator filter $=0\mathrm{.}5kHz,$ calculate the average signal$-$to$-$quantization noise power $($SQNR$)$ of the DM$.$

$(7$ Marks$)$

$($c$)$ The binary information from the DM output of part $($b$)$ is transmitted using $8-$PSK$.$ Assuming that the carrier amplitude is $2$ V and the carrier frequency is $1$ MHz $,$

$($i$)$ determine the time$-$domain expression of all the modulated signals,

$($ii$)$ sketch the corresponding signal$-$space diagram,

$($iii$)$ determine the channel bandwidth required to transmit this signal.

$(8$ Marks$)$