Soru
Q1) Determine whether the following signals are periodic or not . If periodic, specify the fundamental period and-frequency in Hz. a) x(t)=cos^2(2pi t) b) x(t)=cos(t)+sin(2pi t) C) x(t)=e^sint
Çözüm
4.4
(300 Oylar)
Mehtap
Kıdemli · 9 yıl öğretmeni
Uzman doğrulaması
Cevap
To determine whether a signal is periodic, we need to check if there exists a positive constant T, called the period, such that x(t + T) = x(t) for all t. If such a T exists, then the signal is periodic, and T is the fundamental period.a) For x(t) = cos^2(2πt), let's find a T such that x(t + T) = x(t):x(t + T) = cos^2(2π(t + T)) = cos^2(2πt + 2πT)For x(t + T) to equal x(t), we need cos^2(2πt + 2πT) to equal cos^2(2πt). This equality holds true if 2πT is an integer multiple of 2π, i.e., 2πT = 2kπ, where k is an integer. Simplifying, we get T = k.Since T can take any positive value, there exists a T for which x(t + T) = x(t). Therefore, the signal x(t) = cos^2(2πt) is periodic. The fundamental period is T = 1, and the fundamental frequency is f = 1/T = 1 Hz.b) For x(t) = cos(t) + sin(2πt), let's find a T such that x(t + T) = x(t):x(t + T) = cos(t + T) + sin(2π(t + T)) = cos(t) + sin(2πt + 2πT)For x(t + T) to equal x(t), we need cos(t + T) + sin(2πt + 2πT) to equal cos(t) + sin(2πt). This equality holds true if T is an integer multiple of 2π, i.e., 2πT = 2kπ, where k is an integer. Simplifying, we get T = k.Since T can take any positive value, there exists a T for which x(t + T) = x(t). Therefore, the signal x(t) = cos(t) + sin(2πt) is periodic. The fundamental period is T = 2π, and the fundamental frequency is f = 1/T = 1/(2π) Hz.c) For x(t) = e^sin(t), let's find a T such that x(t + T) = x(t):x(t + T) = e^sin(t + T) = e^(sin(t + T))For x(t + T) to equal x(t), we need e^(sin(t + T)) to equal e^sin(t). This equality holds true if sin(t + T) = sin(t) + 2kπ, where k is an integer. Simplifying, we get T = 2kπ.Since T can take any positive value, there exists a T for which x(t + T) = x(t). Therefore, the signal x(t) = e^sin(t) is periodic. The fundamental period is T = 2π, and the fundamental frequency is f = 1/T = 1/(2π) Hz.