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If a continuous-time signal x (t) = cos (2πt) is sampled at 4 Hz, the value of the discrete-time sequence x(n) at n = 5 is
  • a)
    -0.707
  • b)
    -1
  • c)
    0
  • d)
    1
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If a continuous-time signal x (t) = cos (2πt) is sampled at 4 Hz,...
x (t) = cos (2πt)
If the sampling interval is Ts then the sampled signal 
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If a continuous-time signal x (t) = cos (2πt) is sampled at 4 Hz,...
Sampling
Sampling is the process of converting a continuous-time signal into a discrete-time signal by taking samples of the continuous-time signal at regular intervals.

Sampling Formula
The sampling formula is given by:
x(nT) = x(t)|t=nT
where x(nT) is the value of the discrete-time signal at time nT, and x(t) is the value of the continuous-time signal at time t.

Given Signal
The given continuous-time signal is x(t) = cos(2t).

Sampling Rate
The sampling rate is given as 4 Hz.

Sampling Interval
The sampling interval T is 1/4 seconds or 0.25 seconds.

Discrete-Time Sequence
To find the value of the discrete-time sequence x(n) at n = 5, we need to use the sampling formula.

x(5) = x(5T)
x(5T) = x(t)|t=5T
x(5T) = cos(2*5T)
x(5T) = cos(10T)

Value of T
T = 1/4 seconds or 0.25 seconds.

Value of 10T
10T = 2.5 seconds.

Value of cos(10T)
cos(10T) = cos(2.5*4*2*pi)
cos(10T) = cos(20*pi)
cos(10T) = 1

Final Answer
Therefore, the value of the discrete-time sequence x(n) at n = 5 is 1. Hence, option C is the correct answer.
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If a continuous-time signal x (t) = cos (2πt) is sampled at 4 Hz,...
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If a continuous-time signal x (t) = cos (2πt) is sampled at 4 Hz, the value of the discrete-time sequence x(n) at n = 5 isa)-0.707b)-1c)0d)1Correct answer is option 'C'. Can you explain this answer?
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