I need help with my electrical engineering examClass: Communication SystemDate/Time May 12 from 8AM - 9:20 AM EASTERN NY time zone t
ECE454_HW08_SP21_solution SUNY Oswego ECE454: Communications Systems Spring 2021 Homework 8- Solution Problem 1: [10 points] An AM signal ??(??) with a carrier frequency ???? = 100?????? has a complex envelope ??(??) = ????[1 + ??(??)], where ???? = 20??, and the modulation ??(??) is sinusoidal test tone ??(??) = 0.5 sin(2????????), with frequency ???? = 20??????. a) Determine the in-phase and quadrature components ??(??) and ??(??) of this signal. ??(??) = ℜ{??(??)} = ????[1 + ??(??)] ??(??) = ℑ{??(??)} = 0 b) Determine the baseband amplitude and phase ??(??) and ??(??) for this signal. ??(??) = |??(??)| = ????[1 + ??(??)] ??(??) = ∠??(??) = 0 c) Determine the expression of the bandpass signal ??(??). ??(??) = ??(??) cos(?????? + ??(??)) = ????[1 + ??(??)] cos(??????) = 20[1 + 0.5 sin(2????????)] cos(??????) = [20 + 10 sin(2????????)] cos(??????) d) Compute the average normalized power ???? of the AM signal ??(??). The average power of the bandpass signal is ???? = 1 2 ???? The baseband power is ???? = 〈|??(??)|2〉 = 〈|????[1 + ??(??)]|2〉 = ????2〈1 + 2??(??) + ??2(??)〉 Assuming 〈??(??)〉 = 0, then: ???? = ????2 + ????2〈??2(??)〉 = ????2 + ????2〈0.52 sin2(2????????)〉 = ????2 + ????2 4 〈 1 − cos(4????????)) 2 〉 = ????2 + ????2 8 = 9 8???? 2 = 9 8 × 400 = 450?? Therefore, ???? = 1 2 ???? = 225?? e) Using MatLab, plot both the baseband amplitude ??(??) and the bandpass signal v(t). Plot both signals in the same figure over a ??0 = 0.2???? time duration with time resolution (time-step) of ???? = 0.1????. 0 0.5 1 1.5 2 2.5 t 10 -4 -30 -20 -10 0 10 20 30 R(t): baseband envelope v(t): bndpass AM signal Problem 2: [4 points] Using the same AM signal of problem 1, determine the following frequency-domain characteristics: a) Derive the expression of the complex baseband envelope spectrum ??(??). ??(??) = 0.5 sin(2????????) then ??(??) = 0.5 × ?? 2 [??(?? + ????) − ??(?? − ????)] ??(??) = ????[??(??) + ??(??)] = ???? �??(??) + ?? 4 [??(?? + ????) − ??(?? − ????)]� b) Derive the expression of the bandpass spectrum ??(??) using the following equation: ??(??) = 1 2 [??(?? − ????) + ??∗(−?? − ????)] ??(??) = ???? 2 �??(?? − ????) + ?? 4 [??(?? − ???? + ????) − ??(?? − ???? − ????)]� + ???? 2 �??(−?? − ????) + ?? 4 [??(−?? − ???? + ????) − ??(−?? − ???? − ????)]� ∗ = ???? 2 �??(?? − ????) + ?? 4 [??(?? − ???? + ????) − ??(?? − ???? − ????)]� + ???? 2 �??(?? + ????) − ?? 4 [??(?? + ???? − ????) − ??(?? + ???? + ????)]� Problem 3: [6 points] Given a pulse-modulated signal (bandpass) of the form: ??(??) = ??−???? cos[(???? + Δ??)??]??(??) where ??(??) is the unit-step function, a, ωc, and Δω are positive constants and the carrier angular frequency ωc>> Δω. a) Find the corresponding in-phase (I) and quadrature (Q) components x(t) and y(t). You may use the following trigonometric identity: cos(?? + ??) = cos?? ???????? − sin ?? sin ?? ??(??) = ??−???? cos[?????? + Δ????]??(??) ??(??) = ??−???? cos (Δ????)cos(??????)??(??) − ??−???? sin (Δ????)sin(??????)??(??) Or ??(??) = ??−???? cos (Δ????)??(??)cos(??????) − ??−???? sin(Δ????)??(??)sin(??????) = ??(??) cos(??????) − ??(??)sin (??????) where ??(??) = ??−???? cos(Δ????)??(??) and ??(??) = ??−???? sin(Δ????)??(??) are the in-phase (I) and quadrature (Q) components, respectively. b) Determine the magnitude R(t) and phase θ(t) of the complex baseband envelope ??(??). ??(??) = |??(??)| = �??2(??) + ??2(??) = ??−??????(??) ??(??) = arctan � ??(??) ??(??)� = arctan (tan (Δ?? ??)) = Δ?? ?? c) Find the spectrum G(f) of the complex baseband envelope. ??(??) = ??(??)??????(??) = ??(??)????Δ???? Using the Fourier transform properties: ??(??) = ??−??????(??) ↔ 1 ?? + ??2???? ??(??)????Δ???? ↔ 1 ??+??2??�??−Δ??2??� (using frequency-shift) Therefore, ??(??) = 1 ??+??2??�??−Δ??2??� Problem 4: [4 points] An amplifier is tested for total harmonic distortion (THD) by using a single-tone test. The output is observed on a spectrum analyzer. It is found that the peak values of the measured harmonics are ???? = ????????2 � ?? 5 �, where ?? = 1, … ,4. These amplitude are obtained from a single-sided spectrum (?? > 0). - Compute the percentage total harmonic distortion (THD). ?????? = �??22 + ??32 + ??42 ??1 = �????????4 �25�+ ???????? 4 �35�+ ???????? 4 �45� ????????2 �15� = 71.9% n=[1:4]; V=sinc(n/5).^2 THD=sqrt(sum(V(2:4).^2))/V(1) V = 0.8751 0.5728 0.2546 0.0547 THD = 0.7190 ECE454_HW09_SP21_solution SUNY Oswego ECE454: Communications Systems Spring 2021 Homework 9- Solution Problem 1: [8 points] A downconverter circuit is given by the following diagram: The input signal is an AM double-sideband suppressed carrier (DSB-SC) defined as: ??????(??) = ??????(??) cos(2000????) where ??(??) is a real baseband message signal, ???? the carrier frequency and ???? > 0. The input signal is multiplied by a local oscillator (LO) signal ??????(??) = ??0 cos(1600????). a) Determine the carrier frequency ???? and the LO frequency ?????? ???? = 1?????? ?????? = 800???? b) Write an expression of the mixer output ??1(??), assuming that the mixer has a unity gain ???? = 1. What are the two center frequencies at the output of the mixer? Hint: You may use the following identity to convert the product to a sum: cos?? cos?? = 1 2 [cos(?? + ??) + cos(?? − ??)] ??1(??) = ??????(??)??????(??) = ??????(??) cos(2000????)??0 cos(1600????) ??1(??) = ??????0 2 ?? (??)[cos(3600????) + cos(400????)] The two center frequencies at the mixer’s output are 1.8kHz and 200Hz. c) The output filter is a lowpass filter used for frequency down-conversion. Assuming the filter has a gain of 1, write an expression of the output ??2(??) of this system. The filter eliminates the high frequency components of ??1(??). The filter output is: ??2(??) = ??????0 2 ?? (??) cos(400????) d) Write an expression of the spectrum ??2(??) in function of ??(??). ??2(??) = ℱ{??2(??)} = ??????0 4 [??(?? − 200) + ??(?? + 200)] Problem 2: [6 points] An IQ receiver is given by the following diagram: The input signal ??????(??) is a phase modulated (PM) real bandpass signal defined as: ??????(??) = ???? cos(?????? + ??(??)) where ??(??) = 100??(??) is the phase angle and ??(??) is the baseband message signal. a) Determine the outputs ??1(??) and ??2(??) of the two mixers in the receiver above. Write ??1(??) and ??2(??) as a sum of terms. ??1(??) = ??0??????(??) cos(??????) ??1(??) = ??0???? cos(?????? + ??(??)) cos(??????) ??1(??) = ??0???? 2 [cos(2?????? + ??(??)) + cos(??(??))] Similarly, ??2(??) = −??0??????(??) sin(??????) ??2(??) = −??0???? cos(?????? + ??(??)) sin(??????) ??2(??) = −??0???? 2 [sin(2?????? + ??(??)) − sin(??(??))] b) Assuming the low-pass filter (LPF) eliminates the high frequencies of both ??1(??) and ??2(??), write the expressions of both ??3(??) and ??4(??) in function of ??(??). ??3(??) = ??0???? 2 cos (??(??)) = ??0???? 2 cos (100??(??)) ??4(??) = ??0???? 2 sin (??(??)) = ??0???? 2 sin �100??(??)� Problem 3: [5 points] a) Compute the Laplace transform of the phase error θe(t) corresponding to the PLL that is characterized by the following equation: ??????(??) ???? = ??????(??) ???? − ???????????? (??) ∗ ??(??) Express Θ??(??) in function of Θ??(??) and the other system parameters. The Laplace transform of the above equation is: ??Θ??(??) = ??Θ??(??) −????????Θ??(??)??(??) Therefore: �?? + ??????????(??)�Θ??(??) = ??Θ??(??) → Θ??(??) = ?? ??+??????????(??) Θ??(??) b) Using the Laplace transform and the final value theorem, find an expression for the steady-state phase error, lim ??→∞ ????(??), for this PLL. Show that lim??→∞ ????(??) = lim??→∞ ????(??). Hint: The final value theorem islim ??→∞ ??(??) = lim ??→0 ????(??), when the limit exists. lim ??→∞ ????(??) = lim??→0 ??Θ??(??) = lim??→0 ??2 ?? + ??????????(??) Θ??(??) If ??(??) ≠ 0, then lim ??→∞ ????(??) = 0. Therefore, lim??→∞ ????(??) = lim??→∞ ????(??) since ????(??) = ????(??) − ????(??) Problem 4: [4 points] An audio signal with a bandwidth of 8 kHz is transmitted over an AM transmitter with a carrier frequency of 1 MHz. The AM signal is demodulated at the receiver using an envelope detector. a) Draw a block diagram of the envelope detector circuit and label its components. b) What is the constraint on the time constant ?? = ???? for the envelope detector? ?? ≪ 1 2?????? ≪ ???? 1 ???? ≪ 2?????? ≪ 1 ?? 1 2?????? ≪ ???? ≪ 1 2???? 159.15???? ≪ ???? ≪ 19.89???? ECE454_HW10_SP21_solution SUNY Oswego ECE454: Communications Systems Spring 2021 Homework 10- Solution Problem 1: [5 points] A dual-mode cellular phone is designed to operate with either cellular phone service in the 700- MHz band or in the 1,900-MHz band. The phone uses a superheterodyne receiver with a 250- MHz IF for both modes. a) Calculate the LO frequency and the image frequency for high-side injection when the phone receives a 690-MHz signal (from the 700-MHz band). ?????? = ???? + ?????? = 690 + 250 = 940?????? (high-side injection) ???????????? = ?????? + ?????? = 940 + 250 = 1190??????. b) Calculate the LO frequency and the image frequency for low-side injection when the phone receives a 1,940-MHz signal (from the 1900-MHz band). ?????? = ???? − ?????? = 1940− 250 = 1690??????. ???????????? = ?????? − ?????? = 1690− 250 = 1440??????. Problem 2: [8 points] A superheterodyne receiver is tuned to a station at ???? = 40??????. The local oscillator frequency is ?????? = 160?????? and the IF is ?????? = 200?????? (upconversion). a) What is the image frequency ????????????? The system corresponds to an upconverter superheterodyne receiver since ?????? = ???? + ?????? . The image frequency is: ???????????? = ?????? + ?????? = 200 + 160 = 360?????? b) The RF amplifier (i.e. image rejection BPF) has the following frequency response: ??(??) = ?? 1 + ???? �?????? − ??????� where ?? = 100 and ?? = 60. i. Compute the filter magnitude response |??(??)| at both ?? = ???? = 40?????? and ?? = ???????????? . If ?? = 40??????