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A signal x(t) = 100 cos(24π x 103)t is ideally sampled with a sampling period of 50 sec and then passed through an ideal lowpass filter with cutoff frequency of 15 KHz. Which of the following frequencies is/are present at the filter output
- a)12 KHz only
- b)8 KHz only
- c)12 KHz and 9 KHz
- d)12 KHz and 8 KHz
Correct answer is option 'D'. Can you explain this answer?
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A signal x(t) = 100 cos(24π x 103)tis ideally sampled with a sampli...
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A signal x(t) = 100 cos(24π x 103)tis ideally sampled with a sampli...
Sampling the signal:
- The signal x(t) has a frequency of 24 x 10^3 Hz.
- The sampling period is 50 sec, which means the sampling frequency is 1/50 Hz.
- The Nyquist frequency is half of the sampling frequency, which is 1/100 Hz.
- Since the signal frequency (24 x 10^3 Hz) is much higher than the Nyquist frequency (1/100 Hz), the signal is oversampled.
- The sampled signal can be represented as x(nT) = 100 cos(24 x 10^3 nT), where T = 50 sec and n is an integer.
Filtering the signal:
- An ideal lowpass filter with cutoff frequency of 15 KHz is used to filter the sampled signal.
- The transfer function of the filter is H(f) = 1 for f ≤ 15 KHz and H(f) = 0 for f > 15 KHz.
- The filter output can be obtained by convolving the sampled signal with the impulse response of the filter, which is a sinc function.
- The frequency response of the filter can be obtained by taking the Fourier transform of the impulse response.
- The frequency response is a rectangular function with a width of 15 KHz centered at DC.
Frequencies present at the filter output:
- The filter output can be obtained by multiplying the frequency response of the filter with the Fourier transform of the sampled signal.
- The Fourier transform of the sampled signal is a series of impulses at multiples of the sampling frequency (1/50 Hz).
- The frequency response of the filter only allows frequencies up to 15 KHz to pass through.
- Therefore, only the components of the sampled signal that are below 15 KHz will be present at the filter output.
- The frequencies present at the filter output are the frequencies that are common between the Fourier transform of the sampled signal and the frequency response of the filter.
- The frequencies present at the filter output are 12 KHz and 8 KHz, which are both below the cutoff frequency of 15 KHz.
- The correct answer is option 'D'.
- The signal x(t) has a frequency of 24 x 10^3 Hz.
- The sampling period is 50 sec, which means the sampling frequency is 1/50 Hz.
- The Nyquist frequency is half of the sampling frequency, which is 1/100 Hz.
- Since the signal frequency (24 x 10^3 Hz) is much higher than the Nyquist frequency (1/100 Hz), the signal is oversampled.
- The sampled signal can be represented as x(nT) = 100 cos(24 x 10^3 nT), where T = 50 sec and n is an integer.
Filtering the signal:
- An ideal lowpass filter with cutoff frequency of 15 KHz is used to filter the sampled signal.
- The transfer function of the filter is H(f) = 1 for f ≤ 15 KHz and H(f) = 0 for f > 15 KHz.
- The filter output can be obtained by convolving the sampled signal with the impulse response of the filter, which is a sinc function.
- The frequency response of the filter can be obtained by taking the Fourier transform of the impulse response.
- The frequency response is a rectangular function with a width of 15 KHz centered at DC.
Frequencies present at the filter output:
- The filter output can be obtained by multiplying the frequency response of the filter with the Fourier transform of the sampled signal.
- The Fourier transform of the sampled signal is a series of impulses at multiples of the sampling frequency (1/50 Hz).
- The frequency response of the filter only allows frequencies up to 15 KHz to pass through.
- Therefore, only the components of the sampled signal that are below 15 KHz will be present at the filter output.
- The frequencies present at the filter output are the frequencies that are common between the Fourier transform of the sampled signal and the frequency response of the filter.
- The frequencies present at the filter output are 12 KHz and 8 KHz, which are both below the cutoff frequency of 15 KHz.
- The correct answer is option 'D'.
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A signal x(t) = 100 cos(24π x 103)tis ideally sampled with a sampling period of 50sec and then passed through an ideal lowpass filter with cutoff frequency of 15 KHz. Which of the following frequencies is/are present at the filter outputa)12 KHz onlyb)8 KHz onlyc)12 KHz and 9 KHzd)12 KHz and 8 KHzCorrect answer is option 'D'. Can you explain this answer?
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