Question

?h(t) A t +T/2 0 -T/2 Figure 2: Rectangular window time function of width T and height A. (d) Consider a linear time-invariant system (a filter) defined by its impulse response h(t) shown in Figure 2. The filter input is a complex random process x(t) with auto- correlation function $R_x(\tau) = E[x(t)x^*(t - \tau)] = 2N_0\delta(\tau)$, where $N_0$ is a positive constant and $\delta(t)$ is the Dirac delta function. The filter output is denoted by y(t). The input and output power are denoted $P_x$ and $P_y$ respectively. Find $P_x$ and $P_y$.

          ?h(t)
A
t
+T/2
0
-T/2
Figure 2: Rectangular window time function of width T and height A.
(d) Consider a linear time-invariant system (a filter) defined by its impulse response
h(t) shown in Figure 2. The filter input is a complex random process x(t) with auto-
correlation function $R_x(\tau) = E[x(t)x^*(t - \tau)] = 2N_0\delta(\tau)$, where $N_0$ is a positive constant
and $\delta(t)$ is the Dirac delta function. The filter output is denoted by y(t). The input and
output power are denoted $P_x$ and $P_y$ respectively. Find $P_x$ and $P_y$.

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?h(t)
A
t
+T/2
0
-T/2
Figure 2: Rectangular window time function of width T and height A.
(d) Consider a linear time-invariant system (a filter) defined by its impulse response
h(t) shown in Figure 2. The filter input is a complex random process x(t) with auto-
correlation function Rx(τ) = E[x(t)x^*(t - τ)] = 2N0δ(τ), where N0 is a positive constant
and δ(t) is the Dirac delta function. The filter output is denoted by y(t). The input and
output power are denoted Px and Py respectively. Find Px and Py.

Added by Andrea H.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Vishal Gupta

Step 1

The input power is defined as the average power of the input signal x(t). In this case, the input signal x(t) is a complex random process with autocorrelation function Rx(t). To find the input power, we need to integrate the autocorrelation function over all Show more…

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Figure 2: Rectangular window time function of width T and height A. (d) Consider a linear time-invariant system (a filter) defined by its impulse response h(t) shown in Figure 2. The filter input is a complex random process x(t) with autocorrelation function Rx() = E[x(t)x*(t -T)] = 2NoÎ´(t), where No is a positive constant and Î´(t) is the Dirac delta function. The filter output is denoted by y(t). The input and output power are denoted Px and Py, respectively. Find Px and Py.

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