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Two periodic signals x(t) and y(t) have the same fundamental period of 3 seconds. Consider the signal z(t) = x(-t) + y(2t + 1). The fundamental period of z(t) in seconds is
- a)1
- b)1.5
- c)2
- d)3
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Two periodic signals x(t) and y(t) have the same fundamental period o...
The fundamental period of a periodic signal is the smallest positive value of T for which the signal repeats itself. In this problem, we are given two periodic signals x(t) and y(t) with the same fundamental period of 3 seconds. We need to find the fundamental period of the signal z(t) = x(-t) * y(2t + 1).
To find the fundamental period of z(t), we need to find the smallest positive value of T for which z(t + T) = z(t) for all values of t. Let's break down the expression for z(t) and analyze each term separately.
1. x(-t):
The signal x(-t) is a time-reversed version of x(t). If x(t) has a period of T, then x(-t) will also have a period of T. In this case, both x(t) and x(-t) have a fundamental period of 3 seconds.
2. y(2t + 1):
The signal y(2t + 1) is a time-expanded version of y(t) by a factor of 2 and shifted to the left by 1 unit. If y(t) has a period of T, then y(2t + 1) will have a period of T/2. In this case, both y(t) and y(2t + 1) have a fundamental period of 3 seconds.
Now, let's combine both terms to get the expression for z(t):
z(t) = x(-t) * y(2t + 1)
Since both x(-t) and y(2t + 1) have the same fundamental period of 3 seconds, the fundamental period of z(t) will also be 3 seconds. This means that z(t) will repeat itself every 3 seconds.
Therefore, the correct answer is option 'D' - 3 seconds.
To find the fundamental period of z(t), we need to find the smallest positive value of T for which z(t + T) = z(t) for all values of t. Let's break down the expression for z(t) and analyze each term separately.
1. x(-t):
The signal x(-t) is a time-reversed version of x(t). If x(t) has a period of T, then x(-t) will also have a period of T. In this case, both x(t) and x(-t) have a fundamental period of 3 seconds.
2. y(2t + 1):
The signal y(2t + 1) is a time-expanded version of y(t) by a factor of 2 and shifted to the left by 1 unit. If y(t) has a period of T, then y(2t + 1) will have a period of T/2. In this case, both y(t) and y(2t + 1) have a fundamental period of 3 seconds.
Now, let's combine both terms to get the expression for z(t):
z(t) = x(-t) * y(2t + 1)
Since both x(-t) and y(2t + 1) have the same fundamental period of 3 seconds, the fundamental period of z(t) will also be 3 seconds. This means that z(t) will repeat itself every 3 seconds.
Therefore, the correct answer is option 'D' - 3 seconds.
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Community Answer
Two periodic signals x(t) and y(t) have the same fundamental period o...
X(t) → T = 3 seconds
x(-t) → T = 3 seconds
Time period does not change due to time reversal
y(t) → T = 3 seconds
y(t + 1) →T = 3 seconds
Due to shifting time period does not change
y(2t + 1)→ T = 1.5 seconds
Signal will get compress by factor of 2.
Thus x(-t) + y(2t + 1) will have time period
T = LCM (1.5, 3) = 3
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Two periodic signals x(t) and y(t) have the same fundamental period of 3 seconds. Consider the signal z(t) = x(-t) + y(2t + 1). The fundamental period of z(t) in seconds isa)1b)1.5c)2d)3Correct answer is option 'D'. Can you explain this answer?
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