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A sinusoid x(t) of unknown frequency is sampled by an impulse train of period 20 ms. The resulting sample train is next applied to an ideal lowpass filter with cutoff at 25 Hz. The filter output is seen to be a sinusoid of frequency 20 Hz. This means that x(t) is
- a)10 Hz
- b)60 Hz
- c)30 Hz
- d)90 Hz
Correct answer is option 'C'. Can you explain this answer?
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A sinusoid x(t) of unknown frequency is sampled by an impulse train of...
Given Information:
- A sinusoid x(t) of unknown frequency is sampled by an impulse train with a period of 20 ms.
- The resulting sample train is passed through an ideal lowpass filter with a cutoff frequency of 25 Hz.
- The filter output is a sinusoid of frequency 20 Hz.
To Find:
The frequency of the original sinusoid x(t).
Solution:
To solve this problem, we need to understand the concept of sampling and filtering.
1. Sampling:
Sampling is the process of converting a continuous-time signal into a discrete-time signal by taking samples at regular intervals. In this case, the unknown sinusoid x(t) is sampled by an impulse train with a period of 20 ms. Let's call the sampled signal x_s(t).
2. Lowpass Filtering:
After sampling, the sampled signal x_s(t) is passed through an ideal lowpass filter with a cutoff frequency of 25 Hz. The purpose of the lowpass filter is to remove any frequencies higher than the cutoff frequency and retain the frequencies within the passband.
3. Frequency of the Filter Output:
The filter output is observed to be a sinusoid of frequency 20 Hz. This means that the filter is only allowing frequencies up to 20 Hz to pass through and attenuating frequencies higher than 20 Hz. Let's call the filter output y(t).
4. Relationship between Sample Rate and Filter Output Frequency:
The frequency of the filter output y(t) is related to the sample rate and the original frequency of the unknown sinusoid x(t) by the following equation:
Output Frequency = Sample Rate / 2
In this case, the sample rate is the reciprocal of the sampling period, which is 1 / 20 ms = 50 Hz. Therefore, the output frequency is 50 / 2 = 25 Hz.
5. Conclusion:
Since the filter output frequency is 20 Hz, which is lower than the cutoff frequency of 25 Hz, it means that the original frequency of the unknown sinusoid x(t) must be lower than 20 Hz. Therefore, the correct answer is option C - 30 Hz.
To summarize, the unknown sinusoid x(t) has a frequency of 30 Hz based on the given information about sampling, filtering, and the frequency of the filter output.
- A sinusoid x(t) of unknown frequency is sampled by an impulse train with a period of 20 ms.
- The resulting sample train is passed through an ideal lowpass filter with a cutoff frequency of 25 Hz.
- The filter output is a sinusoid of frequency 20 Hz.
To Find:
The frequency of the original sinusoid x(t).
Solution:
To solve this problem, we need to understand the concept of sampling and filtering.
1. Sampling:
Sampling is the process of converting a continuous-time signal into a discrete-time signal by taking samples at regular intervals. In this case, the unknown sinusoid x(t) is sampled by an impulse train with a period of 20 ms. Let's call the sampled signal x_s(t).
2. Lowpass Filtering:
After sampling, the sampled signal x_s(t) is passed through an ideal lowpass filter with a cutoff frequency of 25 Hz. The purpose of the lowpass filter is to remove any frequencies higher than the cutoff frequency and retain the frequencies within the passband.
3. Frequency of the Filter Output:
The filter output is observed to be a sinusoid of frequency 20 Hz. This means that the filter is only allowing frequencies up to 20 Hz to pass through and attenuating frequencies higher than 20 Hz. Let's call the filter output y(t).
4. Relationship between Sample Rate and Filter Output Frequency:
The frequency of the filter output y(t) is related to the sample rate and the original frequency of the unknown sinusoid x(t) by the following equation:
Output Frequency = Sample Rate / 2
In this case, the sample rate is the reciprocal of the sampling period, which is 1 / 20 ms = 50 Hz. Therefore, the output frequency is 50 / 2 = 25 Hz.
5. Conclusion:
Since the filter output frequency is 20 Hz, which is lower than the cutoff frequency of 25 Hz, it means that the original frequency of the unknown sinusoid x(t) must be lower than 20 Hz. Therefore, the correct answer is option C - 30 Hz.
To summarize, the unknown sinusoid x(t) has a frequency of 30 Hz based on the given information about sampling, filtering, and the frequency of the filter output.
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Community Answer
A sinusoid x(t) of unknown frequency is sampled by an impulse train of...
Period of sampling train, Ts = 20 ms
If frequency of x(t) is fx, then after sampling the signal, the sampled signal has the frequency,
fs - fx = 50 - fx and fs + fx = 50 + fs
Now, the sampled signal is applied to and ideal low pass filter with cut off frequency.
fc = 25 Hz
Now, the o/p of filter carried a single frequency component of 20 Hz
∴, only (fs−fx) component passes through the filter, ie.
fs - fx < 25
and fs - fx = 20
50 - fx = 20
fx = 50 - 20 = 30 Hz
If frequency of x(t) is fx, then after sampling the signal, the sampled signal has the frequency,
fs - fx = 50 - fx and fs + fx = 50 + fs
Now, the sampled signal is applied to and ideal low pass filter with cut off frequency.
fc = 25 Hz
Now, the o/p of filter carried a single frequency component of 20 Hz
∴, only (fs−fx) component passes through the filter, ie.
fs - fx < 25
and fs - fx = 20
50 - fx = 20
fx = 50 - 20 = 30 Hz
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A sinusoid x(t) of unknown frequency is sampled by an impulse train of period 20 ms. The resulting sample train is next applied to an ideal lowpass filter with cutoff at 25 Hz. The filter output is seen to be a sinusoid of frequency 20 Hz. This means that x(t) isa)10 Hzb)60 Hzc)30 Hzd)90 HzCorrect answer is option 'C'. Can you explain this answer?
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