- Show that any two statistically independent random variables have a correlation coefficient that is zero.
- Given the random variables\[ \begin{aligned} & U=\cos \Theta \\ & V=\sin \Theta \end{aligned} \]with\[ p_{\Theta}(\theta)= \begin{cases}1 / \pi & -\frac{\pi}{2}
- Prove the following properties of characteristic functions:(a) Every characteristic function has value unity at the origin.(b) The second-order characteristic function \(\mathbf{M}_{U
- A random variable \(U\) is uniformly distributed on the interval \((-A, A)\) and its probability density function is zero otherwise.(a) Find an expression for \(A\) in terms of the variance
- Show that if the moment \(\overline{u^{n} v^{m}}\), if it exists, can be found from the joint characteristic function \(\mathbf{M}\left(\omega_{U}, \omega_{V}\right)\) by the formula\[
- Find the probability density function of the random variable \(Z\) in terms of the known density function \(p_{U}(u)\) when(a) \(z=a u+b\), where \(a\) and \(b\) are real constants.(b)
- Using the method given in Eq. (2.5-25), find the joint probability density function \(p_{W Z}(w, z)\) when\[ \begin{align*} & w=u^{2} \\ & z=u+v \tag{p.2-1} \end{align*} \]and \(p_{U V}(u,
- Consider two independent, identically distributed random variables \(\Theta_{1}\) and \(\Theta_{2}\), each of which obeys a probability density function\[ p_{\Theta}(\theta)=\left\{\begin{array}{cc}
- Consider two identically distributed and independent random variables \(X_{1}\) and \(X_{2}\) with common probability density function \(p_{X}(x)\). Show that the probability density function of the
- Consider the random phasor sum of Section 2.9 with the single change that the phases \(\phi_{k}\) are uniformly distributed on \((-\pi / 2, \pi / 2)\). Find the following quantities: \(\bar{r},
- Let the random variables \(U_{1}\) and \(U_{2}\) be jointly Gaussian, with zero means, equal variances, and correlation coefficient \(ho eq 0\). Consider the new random variables \(V_{1}\) and
- Consider \(n\) independent random variables \(U_{1}, U_{2}, \ldots, U_{n}\), each of which obeys a Cauchy density function,\[ p_{U}(u)=\frac{1}{\pi
- A certain computer contains a random number generator that generates random numbers with uniform relative frequencies (or probability density) on the interval \((0,1)\). Suppose, however, that it is
- Let the real-valued random process U(t)U(t) be defined byU(t)=Acos(2πvt−Φ)U(t)=Acos(2πvt−Φ)where vv is a known constant, ΦΦ is a random variable uniformly distributed on
- Consider the random process \(U(t)=A\), where \(A\) is a random variable uniformly distributed on \((-1,1)\).(a) Sketch some sample functions of this process.(b) Find the time autocorrelation
- An ergodic real-valued random process \(U(t)\) with autocorrelation function \(\Gamma_{U}(\tau)=\left(N_{0} / 2\right) \delta(\tau)\) is applied to the input of a linear, time-invariant filter with
- Consider the random process \(Z(t)=U \cos \pi t\), where \(U\) is a random variable with probability density function\[ p_{U}(u)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{u^{2}}{2}\right) \](a) What
- Find the statistical autocorrelation function of the random process\[ U(t)=a_{1} \cos \left(2 \pi v_{1} t-\Phi_{1}\right)+a_{2} \cos \left(2 \pi v_{2} t-\Phi_{2}\right) \]where \(a_{1}, a_{2},
- A certain random process \(U(t)\) takes on equally probable values +1 or 0 with changes occurring randomly in time. The probability that \(n\) changes occur in time \(\tau\) is known to be\[
- A certain random process \(U(t)\) consists of a sum of (possibly overlapping) pulses of the form \(p\left(t-t_{k}\right)=\operatorname{rect}\left(\left(t-t_{k}\right) / b\right)\) occurring with mean
- Assuming that the random process \(U(t)\) is wide-sense stationary, with mean \(\bar{u}\) and variance \(\sigma^{2}\), which of the following functions represent possible structure functions for
- Prove that the Hilbert transform of the Hilbert transform of a function \(u(t)\) is \(-u(t)\), up to a possible additive constant.
- (a) Show that for an analytic signal representation of a real-valued, narrowband random process, the autocorrelation function of the resulting complex process \(\mathbf{U}(t)\) (assumed wide-sense
- Let \(\mathbf{V}(t)\) be a linearly filtered complex-valued, wide-sense stationary random process with sample functions given by\[ \mathbf{v}(t)=\int_{-\infty}^{\infty} \mathbf{h}(t-\tau)
- Find the power spectral density of a doubly stochastic Poisson impulse process having a rate process \(\Lambda(t)\) described by\[ \Lambda(t)=\lambda_{0}[1+\cos (2 \pi \bar{v} t+\Phi)] \]where
- Starting with Eq. (4.1-10), show that if \(\Delta v \ll \bar{v}\) and \(r \ll 2 c / \Delta v\) for all \(P_{1}\), then\[ \mathbf{u}\left(P_{0}, t\right) \approx \iint_{\Sigma} \frac{e^{j 2 \pi(r /
- (a) Show that the Jones matrix of a polarization analyzer set at angle \(\alpha\) to the \(X\)-axis is given by\[ \underline{\mathbf{L}}(\alpha)=\left[\begin{array}{cc} \cos ^{2} \alpha & \sin
- By finding the trace of the appropriate coherency matrix, show that the average intensity transmitted by a polarization analyzer set at \(+45^{\circ}\) to the \(X\)-axis can be expressed as\[
- By finding the trace of the appropriate coherency matrix, show that the average intensity transmitted by a quarter-wave plate followed by a polarization analyzer set at \(+45^{\circ}\) to the
- Consider a light wave that has \(X\) - and \(Y\)-polarization components of its electric field at point \(P\) given by\[ \begin{aligned} & \mathbf{u}_{X}(t)=\exp \left[-j 2
- Show that the characteristic function of the intensity of polarized thermal light is given by\[ \mathbf{M}_{I}(\omega)=\frac{1}{1-j \omega \bar{I}} \]
- Show that the standard deviation \(\sigma_{I}\) of the instantaneous intensity of partially polarized thermal light is\[ \sigma_{I}=\sqrt{\frac{1+\mathcal{P}^{2}}{2}} \bar{I} \]
- When light falls on a balanced detector (i.e., a detector pair, with one detector output subtracted from the other's output), the output current is proportional to the difference of the intensities
- Let the field emitted by a multimode laser oscillating in \(N\) equals strength and independent modes be represented by\[ \mathbf{u}(t)=\sum_{n=1}^{N} \exp \left[-j\left(2 \pi v_{n}
- Consider a single-mode laser emitting light described by the analytic signal\[ \mathbf{u}(t)=\exp (-j[2 \pi \bar{v} t-\theta(t)]) \](a) Assuming that \(\Delta
- Show that the second moment \(\overline{I^{2}}\) of the intensity of a wave is not equal to the fourth moment \(\overline{\left[u^{(r)}\right]^{4}}\) of the real amplitude of that wave, the
- An idealized model of the normalized power spectral density of a gas laser oscillating in \(N\) equal-intensity axial modes is\[ \widehat{\mathcal{G}}(v)=\frac{1}{N} \sum_{n=-(N-1) / 2}^{(N-1) / 2}
- The gas mixture in a helium-neon laser (end mirrors removed) emits light at \(633 \mathrm{~nm}\) with a Doppler-broadened spectral width of about \(1.5 \times 10^{9} \mathrm{~Hz}\). Calculate the
- (Lloyd's mirror) A point source of narrowband light is placed at distance \(s\) above a perfectly reflecting planar mirror. At distance \(d\) away, the interference fringes are observed on a screen,
- Consider a Young's interference experiment performed with broadband light.(a) Show that the field incident on the observing screen can be expressed as\[ \mathbf{u}(Q, t)=\tilde{K}_{1} \frac{d}{d t}
- As shown in Fig. 5-7p, a positive lens with focal length \(f\) is placed in contact with a pinhole screen in a Young's experiment. The lens and pinhole plane are at distance \(z_{1}\) from the source
- Consider a Michelson interferometer that is used in a Fourier spectroscopy experiment. To obtain high resolution in the computed spectrum, the interferogram must be measured out to large pathlength
- In the Young's interference experiment shown in Fig. 5-5-9pp, the normalized power spectral density \(\widehat{\mathcal{G}}(v)\) of the light is measured at point \(Q\) by a spectrometer. The mutual
- A monochromatic, unit-amplitude plane wave falls normally on a "sandwich" of two diffusers. The diffusers are moving in opposite directions with equal speeds, as shown in Fig. 5-5-10pp. The amplitude
- Show that under quasimonochromatic conditions, mutual intensity \(\mathbf{J}\left(P_{1}, P_{2}\right)\) obeys a pair of Helmholtz equations\[ \begin{aligned} & abla_{1}^{2} \mathbf{J}\left(P_{1},
- Consider an incoherent source radiating with spatial intensity distribution \(I(\xi, \eta)\).(a) Using the Van Cittert-Zernike theorem, show that the coherence area of the light (mean wavelength
- A Young's interference experiment is performed in the geometry shown in Fig. 5-14p. The pinholes are circular and have finite diameter \(\delta\) and spacing \(s\). The source has bandwidth \(\Delta
- The sun subtends an angle of about \(32 \mathrm{~min}\) of arc ( 0.0093 radians) on Earth. Assuming a mean wavelength of \(550 \mathrm{~nm}\), calculate the coherence diameter and coherence area of
- Show that for quasimonochromatic, stationary thermal light, the fourth-order coherence function\[ \boldsymbol{\Gamma}_{1234}\left(t_{1}, t_{2}, t_{3}, t_{4}\right)=E\left[\mathbf{u}\left(P_{1},
- The output of a single-mode, well-stabilized laser is passed through a spatially distributed phase modulator (or a phase-only spatial light modulator that is changing with time). The field observed
- Show that for light with a Lorentzian spectral profile, the parameter \(\mathcal{M}\) is given by\[ \mathcal{M}=\left[\frac{\tau_{c}}{T}+\frac{1}{2}\left(\frac{\tau_{c}}{T}\right)^{2}\left(e^{-2 T /
- Examination of Fig. 6.5 shows that a relatively abrupt threshold in values of \(\tilde{\lambda}_{n}\) occurs as \(n\) is varied. In particular, as a rough approximation,\[ \tilde{\lambda}_{n} \approx
- Suppose that we wish to know the standard deviation of the phase of \(\mathbf{J}_{12}(T)\) under the condition that the measurement time \(T\) is sufficiently long compared with the correlation time
- Under the same conditions described in the previous problem, the fluctuations of the length of \(\mathbf{J}_{12}(T)\) are caused mainly by the fluctuations in the real part of the noise. Making this
- Compare the rms signal-to-(self) noise ratios achievable using amplitude interferometry and intensity interferometry to measure \(\mu_{12}\) under the assumption that \(T / \tau_{c} \gg 1 /
- Find what modification of Eq. (6.3-32) must be made if the light incident on the detectors is unpolarized thermal light. SM rms M/2 1+2 B/b - b 4B (6.3-32)
- Given that the condenser lens of Fig. 7.4 is circular with diameter \(D\), specify the diameter required of a circular incoherent source to assure that the approximation of Eq. (7.2-7) is valid. (a,)
- In the optical system shown in Fig. 7-4p, a square incoherent source ( \(L \times L\) meters) lies in the source plane. The object consists of two pinholes separated in the \(\xi\) direction by
- A partially polarized thermal light wave is incident on a photodetector. The total incident integrated intensity can be regarded to consist of two statistically independent components, \(W_{1}\)
- Light from a partially polarized pseudothermal source is found to have a coherence matrix of the form\[ \underline{\mathbf{J}}=\bar{I}\left[\begin{array}{ll} 1 / 2 & -1 / 6 \\ -1 / 6 & 1 / 2
- Assuming factorability of the complex degree of coherence of polarized thermal light, demonstrate that when the photodetector area is much larger than the coherence area of the incident light, the
- Given the assumption that the energy levels of an harmonic oscillator can take on only the values \(n h v\) and given the Maxwell-Boltzmann distribution of occupation numbers, show that the
- Show that when two identical detectors in an intensity interferometer have read noise with the same variance \(\sigma_{\text {read }}^{2}\) and when the average intensities incident on the two