Question

(TCO 4) A signal consists of only two sinusoids, one of 65 Hz and one of 95 Hz. This signal is sampled at a rate of 245 Hz. Find the first six positive frequencies that will be present in the replicated spectrum.

Answer 1

65 Hz, 95 Hz, 150 Hz, 180 Hz, 310 Hz, 340 Hz

Step-by-step explanation:

Given :

Frequencies of the sinusoids,

`$f_(m_1)= 65 \ Hz$` , and

`$f_(m_2)= 95 \ Hz$`

Sampling rate
`f_s = \ 245 \ Hz`

The positive frequencies at the output of the sampling system are :

`$f_(o_1)=\pm f_(m_1) \pm nf_s, f_(o_2)=\pm f_(m_2) \pm nf_s $`

When n = 0,

`$f_(o_1)= f_(m_1) = 65 \ Hz,\ \ f_(o_2)= f_(m_2) = 95 \ Hz $`

when n = 1,

`$f_(o_1)=\pm f_(m_1) \pm f_s, \ \ f_(o_2)=\pm f_(m_2) \pm f_s $`

`$f_(o_1)= \pm 65 \pm 245,\ \ f_(o_2)=\pm 95 \pm 245$`

`$f_(o_1)= 180 \ Hz, 310 \ Hz,\ \ f_(o_2)= 150 \ Hz,340 \ Hz$`

When n = 2,

`$f_(o_1)= \pm 65 \pm 2(245),\ \ f_(o_2)=\pm 95 \pm 2(245)$`

`$f_(o_1)= 555 \ Hz, 425 \ Hz,\ \ f_(o_2)= 395 \ Hz,585 \ Hz$`

Therefore, the first six positive frequencies present in the replicated spectrum are :

65 Hz, 95 Hz, 150 Hz, 180 Hz, 310 Hz, 340 Hz