EE 511 Problem Set 7

EE 511 Problem Set 7

Due on 16 Nov 2007

1. Let ˆX_t be the Hilbert transform of the W.S.S. random process X_t. Show that (i) S_Xˆ(f) = SX(f), (ii) S_XXˆ(f) = −S_XXˆ (f), (iii) E[XtXˆt] = 0, and (iv) for Zt =Xt+jXˆt, determine S_Z(f) in terms ofS_X(f).

2. The power spectrum of a W.S.S. band-pass random processXtis as shown in Figure 1. Sketch S_XI(f),S_XQ(f) andS_XIX^Q(f) assuming 5 MHz to be the carrier frequency.

S (f)X

f(MHz)

4 5 7

−7 −5 −4

1

Figure 1:

3. Repeat problem 2 assuming 4 MHz to be the carrier frequency.

4. A narrow-band noise process Nt has zero-mean and auto-correlation function RN(τ). Its power spectral density S_N(f) is centered about ±f_c. The in-phase and quadrature compo- nentsN_t^I and N_t^Q are defined byN_t^I =N_tcos 2πfct+ ˆN_tsin 2πfct and N_t^Q = ˆN_tcos 2πfct− Ntsin 2πfct. Show thatR_NI(τ) =R_NQ(τ) =RN(τ) cos 2πfcτ+ ˆRN(τ) sin 2πfcτ andR_NIN^Q(τ) =

−R_NQN^I(τ) =R_N(τ) sin 2πf_cτ −Rˆ_N(τ) cos 2πf_cτ, where ˆR_N(τ) is the Hilbert transform of R_N(τ).

5. A pair of noise processesN_tandW_tare related byN_t=W_tcos (2πf_ct+ Θ)−W_tsin (2πf_ct+ Θ), wheref_c is a constant and Θ is a uniform random variable in the interval [0,2π]. The noise processWtis stationary and its power spectral density is as shown in Figure 2. Find and plot the power spectral density ofN_t,S_N(f).

S (f) W

f a

W

−W 0

Figure 2:

6. A W. S. S. band-pass random process X_t is generated from two W. S. S. low-pass random processes X_t^I and X_t^Q as follows: X_t =X_t^Icos 2πfct−X_t^Qsin 2πfct. S_XI(f) and S_XIX^Q(f) are as shown in Figure 3. (a) SketchS_X^Q(f) andS_X^Q_X^I(f). (b) Determine and sketchSX(f)

assumingfc = 10M Hz. (c) Determine and sketch S_X˜(f), the power spectral density of the complex baseband equivalent process given byX_t^I+jX_t^Q.

S (f)

f (MHz) 1 f(MHz)

1

−1 1

−4

XI

4

−1

−4 4

X XI Q(f)

X XI Q(f) Im(S )

Re(S ) = 0

Figure 3:

7. The power spectral density of a wide-sense stationary band-pass random process Xt is as shown in Figure 4.

10

−14

1 S (f)

X

f (MHz)

−10 0

−12 12 14

Figure 4:

(a) Assuming that the carrier frequency is 11 MHz, sketchS_X^I(f), the power spectral density of the in-phase component of the band-pass process, and mark the important details.

(b) Assuming that the carrier frequency is 11 MHz, sketchS_XIX^Q(f), the cross power spectral density of the in-phase and quadrature components of the band-pass process, and mark the important details.

(c) How areS_XQ(f) andS_XQX^I(f) related toS_XI(f) andS_XIX^Q(f)?

2