A signal x(t)=100cos(24π×103t) is ideally sampled with a sampling period of Ts=50 µs...

 

Question 36

A signal

x(t)=100cos(24π×10³t)

is ideally sampled with a sampling period of

T_s=50 µs

and then passed through an ideal low-pass filter having a cutoff frequency of 15 kHz. Which frequencies are present at the filter output?

1. 12 kHz only
2. 8 kHz only
3. 12 kHz and 9 kHz
4. 12 kHz and 8 kHz

Solution

Step 1: Find the signal frequency

Since

2πf=24π×10³

Therefore,

f=12 kHz

Step 2: Find the sampling frequency

f_s=1/T_s

=1/(50×10^-6) =20000 Hz =20 kHz

Step 3: Spectral Replicas

The sampled spectrum contains frequencies

|nf_s±f|

For n = 0

12 kHz

For n = 1

20-12=8 kHz

20+12=32 kHz

For n = 2

40-12=28 kHz

and so on.

Step 4: Apply LPF

LPF cutoff frequency

15 kHz

The frequencies below 15 kHz are

• 8 kHz ✔
• 12 kHz ✔

Therefore, both components pass through the filter.

Final Output Frequencies

8 kHz and 12 kHz

Correct Answer: (d)

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