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Consider two real valued signals, x(t) band-limited to [−500 Hz, 500Hz] and y(t) bandlimited to [−1kHz, 1kHz]. For z (t) = x(t). y(t), the Nyquist sampling frequency (in kHz) is __________
- a)2
- b)3
- c)4
- d)6
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Consider two real valued signals, x(t) band-limited to [−500 Hz, 500H...
To determine the Nyquist sampling frequency for the signal z(t) = x(t) * y(t), we need to consider the bandwidths of both x(t) and y(t).
Given:
- x(t) is band-limited to [-500 Hz, 500 Hz]
- y(t) is band-limited to [-1 kHz, 1 kHz]
Understanding Bandwidths:
Bandwidth is defined as the range of frequencies within which a signal is significant or non-negligible. In this case, the band-limited ranges specify the frequencies for which the signals x(t) and y(t) have significant energy.
Nyquist Sampling Theorem:
According to the Nyquist-Shannon sampling theorem, a continuous-time signal must be sampled at a rate greater than or equal to twice its bandwidth to avoid aliasing. Mathematically, the Nyquist sampling frequency (Fs) is given by Fs = 2 * B, where B is the bandwidth of the signal.
Determining the Nyquist Sampling Frequency for z(t):
To find the Nyquist sampling frequency for z(t), we first need to determine the combined bandwidth of x(t) and y(t) when multiplied together.
- The bandwidth of x(t) is 500 Hz (from -500 Hz to 500 Hz).
- The bandwidth of y(t) is 1 kHz (from -1 kHz to 1 kHz).
When two signals are multiplied together in the time domain, their spectra are convolved in the frequency domain. This means that the bandwidth of the resulting signal is the sum of the individual bandwidths.
- The bandwidth of z(t) = x(t) * y(t) is 500 Hz + 1 kHz = 1.5 kHz.
Finally, we can determine the Nyquist sampling frequency for z(t) by multiplying its bandwidth by 2.
- Nyquist sampling frequency for z(t) = 2 * 1.5 kHz = 3 kHz.
Therefore, the correct answer is option 'B' (3 kHz).
Given:
- x(t) is band-limited to [-500 Hz, 500 Hz]
- y(t) is band-limited to [-1 kHz, 1 kHz]
Understanding Bandwidths:
Bandwidth is defined as the range of frequencies within which a signal is significant or non-negligible. In this case, the band-limited ranges specify the frequencies for which the signals x(t) and y(t) have significant energy.
Nyquist Sampling Theorem:
According to the Nyquist-Shannon sampling theorem, a continuous-time signal must be sampled at a rate greater than or equal to twice its bandwidth to avoid aliasing. Mathematically, the Nyquist sampling frequency (Fs) is given by Fs = 2 * B, where B is the bandwidth of the signal.
Determining the Nyquist Sampling Frequency for z(t):
To find the Nyquist sampling frequency for z(t), we first need to determine the combined bandwidth of x(t) and y(t) when multiplied together.
- The bandwidth of x(t) is 500 Hz (from -500 Hz to 500 Hz).
- The bandwidth of y(t) is 1 kHz (from -1 kHz to 1 kHz).
When two signals are multiplied together in the time domain, their spectra are convolved in the frequency domain. This means that the bandwidth of the resulting signal is the sum of the individual bandwidths.
- The bandwidth of z(t) = x(t) * y(t) is 500 Hz + 1 kHz = 1.5 kHz.
Finally, we can determine the Nyquist sampling frequency for z(t) by multiplying its bandwidth by 2.
- Nyquist sampling frequency for z(t) = 2 * 1.5 kHz = 3 kHz.
Therefore, the correct answer is option 'B' (3 kHz).
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Community Answer
Consider two real valued signals, x(t) band-limited to [−500 Hz, 500H...
X(t ) is band limited to [−500Hz, 500Hz] y(t )is band limited to [−1000Hz, 1000Hz]
z(t ) = x (t).y(t)
Multiplication in time domain results convolution in frequency domain.
The range of convolution in frequency domain is [−1500Hz, 1500Hz]
So maximum frequency present in z(t) is 1500Hz Nyquist rate is 3000Hz or 3 kHz
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