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#### Let the signals x(t) and y(t) be the input and output signals to a differentiator, respectively....

Let the signals

*x*(*t*) and y(*t*) be the input and output signals to a differentiator, respectively.*x*(*t*) do y(*t*) (a) Let the*autocorrelation*of the*signal**x*(*t*) be*R*(*T*) and the*autocorrelation*of the*signal*y(*t*) be*R*(*T*). If y(*t*)=*X*, express*R*, (*T*) in terms of*R*. (*T*) dt (b) Assume*R*(*T*) = 5e and find...#### 151 4.8 Problems 4.17 The autocorrelation function Txx(t) of a real analog signal x (t) is defined by xx(t) x(t)x...

151 4.8 Problems 4.17 The

*autocorrelation**function*Txx(*t*) of a real analog*signal**x*(*t*) is defined by xx(*t*)*x*(*t*)*x*(*r*-*t*) dt, (4.73) obtained from Eq. (4.70) by replacing y(*t*) with*x*(*t*). Thus the*function*is a cross-correlation of*x*(*t*) with itself. One application of the*autocorrelation**function*is to detect the period of a periodic*signal*that has...#### 1.12. The Fourier transform of a signal x(t) is defined by X(f) = sincf, where the sinc func- tion is as defined in Equ...

1.12. The Fourier transform of a

*signal**x*(*t*) is defined by*X*(f) = sincf, where the sinc func- tion is as defined in Equation (1.39). Find the*autocorrelation**function*,*R*.(*T*), of the*signal**x*(*t*). 1.12. The Fourier transform of a*signal**x*(*t*) is defined by*X*(f) = sincf, where the sinc func- tion is as defined in Equation (1.39). Find...#### Consider a sinusoidal signal with random phase, defined by x(t)=Acos(2πfct+θ), where A and FC are constant and θ is a random variable uniform distributed over interval [-π,π], that is f(θ)={█(1/2π,&-π≤θ≤π@0,&elsewhere)┤ Describe the autocorrelation RX(τ

*Consider*a sinusoidal*signal*with random phase, defined by , where A and FC are constant and is a random variable uniform distributed over interval [-, that isa) Describe the*autocorrelation*RX of a sinusoidal wave*X*(*t*)b) Describe the*power*spectral density SX of a sinusoidal wave*X*(*t*)*Consider*a sinusoidal*signal*with random phase, defined by , where A and...#### Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t)...

*Consider*the RC circuit shown below. Assume that*R*=(0.1)2 and C=(0.1)F 3.*R*i(*t*) y (*t*)*x*(*t*) The input to this circuit is given as*x*(*t*) s(*t*)+ny (*t*), where the noise component of input, n(*t*), is a sample*function*realization of white noise process with an*autocorrelation**function*given by Rpx(*t*) 8(*T*), and s (*t*) cos(6Tt) is the*signal*component...#### Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t)...

*Consider*the RC circuit shown below. Assume that*R*=(0.1)2 and C=(0.1)F 3.*R*i(*t*) y (*t*)*x*(*t*) The input to this circuit is given as*x*(*t*) s(*t*)+ny (*t*), where the noise component of input, n(*t*), is a sample*function*realization of white noise process with an*autocorrelation**function*given by Rpx(*t*) 8(*T*), and s (*t*) cos(6Tt) is the*signal*component...#### 9. Random binary signal x(t) transmit one digit every Tb seconds. A binary bit '1, is transmitted...

9. Random binary

*signal**x*(*t*) transmit one digit every Tb seconds. A binary bit '1, is transmitted by a rectangular pulse of width*T*&/2 and amplitude of 1. A binary bit 'O' is transmitted by no*signal*. The digits1 and '0' are equal likely and occur randomly. Determine the*autocorrelation**function*and the*power*spectrum density. (20 points) 3 9....#### 4. Find and the autocorrelation function R (t) of the following signal : x( Find the...

4. Find and the

*autocorrelation**function**R*(*t*) of the following*signal*:*x*( Find the energy Ex#### 1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay...

1) Random Processes: Suppose that a wide-sense stationary Gaussian random process

*X*(*t*) is input to the filter shown below. The*autocorrelation**function*of*X*(*t*) is 2xx (*r*) = exp(-ary Y(*t*)*X*(*t*) Delay a) (4 points) Find the*power*spectral density of the output random process y(*t*), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c)...#### 3. A white Gaussian noise signal W (t) with autocorrelation function it passes through a linear filter invariant in time h (t). Calculate the average power of the exit process Y (t) knowing that We w...

3. A white Gaussian noise

*signal*W (*t*) with*autocorrelation**function*it passes through a linear filter invariant in time h (*t*). Calculate the average*power*of the exit process Y (*t*) knowing that We were unable to transcribe this image*r*. h2 (*t*)dt = 1.*r*. h2 (*t*)dt = 1.

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