Communications Engineering Task Collection

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1 Communications Engineering ask Collection International Graduate Program Master of Science in Communications and Signal Processing Dr. Mike Wolf, Communications Research Laboratory (Fachgebiet Nachrichtentechnik) last modified: January 26,

2 1 System theoretic basics Bandpass lowpass transformation 1.1 Determine the complex envelopes u L1 (t) and u L2 (t) for the following (real valued) bandpass signals (a) u 1 (t) = U 0 cos(2πf c t), (b) u 2 (t) = U 0 sin(2πf c t). he transformation is done with respect to the frequency f c. 1.2 he following figure shows the spectrum X(f) of a (real valued) bandpass signal. (a) Draw the spectrum X L (f) of the corresponding baseband signal, if the bandpass lowpass transformation will be performed regarding the frequency f c. (b) Determine the corresponding signal x L (t) in the time domain. (c) Specify the inphase as well as the quadrature component of x(t)? X(f) X 0 f c f c f X 0 2B 1.3 he following figure shows the spectrum X(f) of a (real valued) bandpass signal. (a) Draw the spectrum X L (f) of the corresponding lowpass signal, if the bandpass lowpass transformation will be performed regarding the frequency f c. (b) Determine the corresponding signal x L (t) in the time domain. X(f) X 0 f c f c f B 2

3 1.4 he following complex baseband signal is considered x L (t) = j X 0 si(πbt). (a) Calculate the corresponding bandpass signal x(t), if the frequency f c = 5 B is supposed for the transformation from the lowpass range into the bandpass range. (b) Determine the spectrum X(f) of x(t). 1.5 he following amplitude modulated signal u 1 (t) = U 0 cos(2πf s t)sin(2πf c t) with f c f s will be bandpass filtered, where the filter transfer function is given as ( ) ( ) f + fc + B/2 f fc B/2 G(f) = rect + rect with B = 2f s. B B (a) Determine the signal u 2 (t) and the corresponding spectrum U 2 (f) at the bandpass filter output (equation). (b) Make a bandpass lowpass transformation with respect to the frequency f c. Draw the input (baseband) signal u L1 (t) and the corresponding spectrum U L1 (f). (c) Draw the impulse response g L (t) and the transfer function G L (f) in the equivalent baseband range. Please note that the transformation is still done with respect to the frequency f c. (d) Determine the output signal u L2 (t) and its spectrum U L2 (f) in the equivalent baseband range (equation and figure). 3

4 Correlation functions of deterministic signals 1.6 Consider the following two signals ( t /2 u 1 (t) = U 0 rect ), u 2 (t) = U 0 [ rect ( t /4 /2 ) rect ( t 3/4 (a) Determine the auto-correlation functions ϕ E 11 (τ) and ϕe 22 (τ) (figures). (b) Determine the according energy spectral densities Φ E 11 (f) and ΦE 22 (f) (figures). (c) Determine the cross-correlation function ϕ E 12 (τ) (figure). 1.7 Consider the following two signals /2 u 1 (t) = U 1 cos (2πµf 0 t + φ 1 ), u 2 (t) = U 2 cos (2πνf 0 t + φ 2 ), µ,ν N. Both signals have the common period t p = 1/f 0, which should be used to calculate the correlation functions. (a) Calculate the average power for each of the signals. )]. (b) Determine the auto-correlation functions ϕ 11 (τ) and ϕ 22 (τ) (figures and equations). Note: cos(x) cos(y) = 1 2 (cos(x y) + cos(x + y)). (c) Determine the cross-correlation function ϕ 12 (τ) for µ ν (equation). (d) Determine the auto-correlation function of the sum signal u s (t) = u 1 (t) + u 2 (t) and the corresponding power spectral density for µ ν (figures and equations). (e) Derive the average power of u s (t) from its spectral density. (f) Determine the autocorrelation function of the real valued signal u p (t) = µ= C µ exp(j2πµf 0 t), where C µ = C µ. 1.8 Consider the following periodic signal u p (t) = u(t nt p ) with ( ) t /2 u(t) = U 0 rect and t p = 4. (a) Determine the power spectral density of u p (t) (figure and equation) (b) Determine the auto-correlation function of u p (t) (figure and equation). 4

5 Correlation functions of stochastic signals 1.9 he figure shows a binary stochastic signal. Pulses and pulse breaks have the same probability. A pulse as a pulse break extents over an interval p. he occurrence of a pulse or a pulse break does not depend on the previous signal run. (a) Determine the first moment m (1) s of s(t). Compare ensemble averaging and averaging in time. (b) Determine the second moment m (2) s of s(t). (c) Determine the second central moment µ (2) s. Specify the relations between mean power, AC-power and DC-power. (d) Determine the probability density function f s (x) of s(t). (e) Determine the joint probability density function f sts t+τ (x,y,τ) of the processes s(t) and s(t + τ) for τ = 0, τ p and τ = p /2. (f) Determine the auto-correlation function ϕ ss (τ) (figure and equation). (g) Determine the power spectral density Φ ss (f) (figure and equation). s(t) U 0 p 1.10 We consider an OOK bandpass signal in the equivalent lowpass range. he equivalent lowpass signal is given as ( ) 2 t s L (t) = ( 1) n b /4 b n 2Eb ψ L (t n ) with ψ L (t) = rect. /2 denotes the bit interval; b n {0,1} are the bits to be transmitted. he states 0 and 1 have the same probability, where subsequent bits are independent. In this example, the pulse width is /2. Please take into account that all bits with an odd index are weighted with 1. (a) Draw the signal for the bit sequence {b 0,b 1,b 2,b 3,b 4,b 5,b 6,b 7,b 8,b 9 } = {1,1,0,1,0,0,1,1,1,0} in the interval 0 t 10. (b) Determine the first moment m (1) s of s(t). (Please consider a time interval of infinite size.) (c) Denote the mean power m (2) s and the variance (AC-power) µ (2) s depending on E b. 5 t

6 (d) Draw the probability density function f s (x) of s(t). (e) Draw the auto-correlation function ϕ ss (τ) of s(t). (f) Draw the power spectral density in the lowpass range as well as in the bandpass range (a) Determine the auto-correlation function ϕ pp (τ) as well as the power spectral density Φ pp (f) of a periodic rectangular pulse train according to u p (t) = 1 t Discuss the limit t 0. Note: use averaging in time ( ) t ntp rect t with t t p. (b) Consider an ensemble of periodic rectangular pulse trains. It is assumed that each realization k u p (t) = 1 ( ) t ntp t k rect. t t (index k ) has a random delay t k, where t k is uniformly distributed within [0;t p ). Determine the autocorrelation function E{U p (t)u p (t + τ)} (ensemble averaging). (c) he NRZ On-Off Keying (OOK) signal s(t) = can be also written as ( ) 1 t b /2 b n 2Eb ψ(t n ) with ψ(t) = rect s(t) = ψ(t) 2E b b n δ(t n ), } {{ } s D (t) where b n {0,1} are the bits to be transmitted. he states 0 and 1 have the same probability, where subsequent bits are independent. Determine the autocorrelation functions of ψ(t), s D (t) and s(t). Since s(t) and s D (t) are cyclo-stationary, we assume a derived random process for ensemble averaging, where each realization is supposed to have an additional delay uniformly distributed within [0; ). 6

7 2 Optimal filters Matched filter 2.1 Consider the energy signal ( ) t /4 u 1A (t) = U 0 rect U 0 rect /2 which is corrupted by additive white Gaussian noise. ( t 3/4 (a) Determine the impulse response and the transfer function of a causal matched filter. /2 (b) Determine the response of the filter, if the input signal is u 1A (t). (c) Determine the signal-to-noise ratio η at the sampling time. (d) Determine the response of the filter, if the input signal is u 1B (t) = U 0 rect 2.2 Consider the bandpass signal x(t) = U 0 rect which is corrupted by white noise. ), ( ) t sin(2πf c t) mit f c 1, (a) Draw x(t) and its amplitude spectral density X(f). (b) Determine the energy E x of x(t)? ( ) t /2 (c) Determine the equivalent lowpass signals x L (t) and X L (f), where the bandpass lowpass transformation is done with respect to f c (figures). (d) Determine the impulse response g(t) of a unit energy matched filter. he optimal sampling time is supposed to be t = 0. (e) Determine the impulse response g L (t) in the equivalent lowpass range. (f) Determine the filter output signal, if the input signal is x(t) and x L (t), respectively. (g) Determine the SNR η at the sampling time both in the bandpass and in the equivalent lowpass range. 7

8 3 Signal Space 3.1 Consider the following 4 energy signals s i (t). s 1 (t) = { 1 V for 0 t < /3 0 V else { 1 V for 0 t < 2/3, s 2 (t) = 0 V else, s 3 (t) = { 1 V for /3 t < 0 V else { 1 V for 0 t < and s 4 (t) = 0 V else. All 4 signals are limited to the time interval [0,]. (a) Show that the signals do not built a set of orthogonal functions. (b) Find a set of orthogonal basis function. Use the Gram-Schmidt orthogonalization procedure. (c) Draw the signal space diagram. 3.2 Consider the following 5 signals. For 0 t < the signals are defined as s 1 (t) = 2U 0 cos(2πf c t); s 2 (t) = 2U 0 cos(2πf c t); s 3 (t) = 2U 0 sin(2πf c t); s 4 (t) = 2 U 0 cos(2πf c t + π/4); s 5 (t) = 2 U 0 cos(2πf c t + 3π/4). Outside the time interval [0,] all signals disappear. he center frequency f c is assumed to be f c = k 1/, where k 0, integer. (a) Determine the dimension N of the signal space. (b) Determine the basis functions ψ j (t), j = 1,...,N, according to the Gram-Schmidt orthogonalization procedure. (c) Determine the signal vectors s i, i = 1,...,5. (d) Sketch the signal space diagram. (e) Calculate the signal energies E i, i = 1,...,5. addition theorems: cos(x + y) = cos(x)cos(y) sin(x)sin(y); cos 2 (x) = 1 [1 + cos(2x)]; 2 sin(x)cos(y) = 1 2 [sin(x y)+sin(x+y)]; cos(x)cos(y) = 1 2 [cos(x y)+cos(x+y)] 8

9 4 Discrete Modulation 4.1 We consider a binary phase-shift keying (BPSK) signal, which is equipped with an additional unmodulated sin( )-carrier for Rx-synchronization. For 0 t < we can write { s1 (t) = ku u tx (t) = 0 sin(2πf c t) + 1 k 2 U 0 cos(2πf c t) for b 0 = 1, s 2 (t) = ku 0 sin(2πf c t) 1 k 2 U 0 cos(2πf c t) for b 0 = 0, where k, 0 k < 1, is a constant which determines the relative power of the unmodulated sin( )-carrier. (a) Sketch the signal space diagram. (b) Show that the BER p b is given by p b = 1 2 erfc (1 k 2 )E b N 0. (c) Compare the required power of this scheme to conventional BPSK. 4.2 Consider the coherent detection of 2-PSK, if the receiver uses a correlation type detector which exhibits a phase error φ e. Discuss the impact of φ e on the BER p b. 4.3 Consider the following On-Off Keying signal u tx (t) = 2E b b n ψ 1 (t n ) with ψ 1 (t) = 1 rect ( t b /2 (a) Determine the power spectral density of u tx (t). It is assumed that adjacent bits b n {0,1} are independent. Both states of b n have the same probability. (b) Derive the BER p b for coherent detection in AWGN, where the physical noise power spectral density is N Determine the noise power spectral density of a pseudo-ternary signal according to u tx (t) = ( ) 1 t 2E b b n ( 1) n b /2 ψ 1 (t n ) with ψ 1 (t) = rect. Compared to conventional OOK, bit with an odd index are weighted with 1. (It is assumed that adjacent bits b n {0,1} are independent. Both states of b n have the same probability.) 4.5 Determine the noise power spectral density of a pseudo-ternary signal according to u tx (t) = ( ) 1 t b /2 2E b z n ψ 1 (t n ) with ψ 1 (t) = rect. z n are AMI-coded pseudo-ternary symbols, which are given as z n = b n b n 1 with b n = b n b n 1. he symbols b n are differentially encoded bits, where it is assumed that adjacent bits bn {0,1} are still independent. Both states of b n have the same probability. ). 9

10 4.6 he following figure shows two orthogonal signals s 1 (t) and s 2 (t), which are used for binary modulation. he bitrate is R b = 1/. s 1 (t) and s 2 (t) are generated with the same probability. If only the first bit interval is considered, the received signal is y(t) = s i (t) + n(t), where n(t) is AWGN with the technical power spectral density N 0. (a) How does the optimal detector look like? (b) Determine the BER p b for this case. s 1 (t) s 2 (t) U 0 U 0 0 t 0 t U 0 U Determine the BER p b (depending on E b /N 0 ) for 4-PSK detection in AWGN. Assume that the modulator uses quadrature modulation with two independent 2-PSK data streams. 4.8 Determine the symbol error rate p s as well as the BER p b for bipolar 16-ASK detection in AWGN. Compare the results with 16-QAM detection. Note: (M 1) 2 = M 2 M Determine the power spectral density of M PSK. It is assumed that the transmitter uses a baseband pulse shaping filter with the impulse response g ps (t). he transmitted bandpass signal is u tx (t) = s zn+1(t n), where z n {0,1,2,...,M 1} are the M-ary symbols to be transmitted. = ldm is the symbol interval. he elementary signals are given as ( s i (t) = U 0 cos 2πf c t + (i 1) 2π ) g ps (t) i = 1,2,...,M. M (a) Determine the elementary signals in the equivalent baseband for ( ) t /2 g ps (t) = rect. (b) Determine the equation of the transmitted signal in the equivalent baseband. 10

11 (c) Determine its power spectral density. (d) Determine the power spectral density of the bandpass signal Show that the bit error rate of binary transmission in AWGN is given as p b = 1 ( ) 2 erfc dmin 2, N 0 if ideal coherent detection is used. N 0 is the (technical) power spectral density; d min denotes the Euclidian distance d(s 1,s 2 ) between the 2 dimensional signal vectors. Note: Assuming that s 1 was transmitted, an error occurs if y s 1 > y s 2. y denotes the receive vector An Offset 4-QAM (QPSK) signal, given in the equivalent baseband, can be written as u tx,l (t) = E b k= d k ψ 0(t k) + j E b k= d k ψ 0(t k /2). is the symbol interval. Both d k and d k are chosen from the alphabet { 1,+1}. (a) Determine the autocorrelation function ϕ uu,l (τ) of u tx,l as a function of the impulse response ψ 0 (t) of the pulse shaping filter. (b) Determine the corresponding power spectral density. (c) Minimum-Shift Keying (MSK) is be considered as Offset QPSK with the following pulse shape { 2 ψ 0 (t) = cos ( π t ) for 2 t < 2. 0 else. Determine the power spectral density of MSK. Note: Use the following relation: { ( ) cos 2π t u(t) = t p for tp 4 t < tp 4 0 else U(f) = t p π cos ( π 2 t pf ) 1 (t p f) he transmit signal of MSK is defined in the previous task. he signal is given in the equivalent baseband range. (a) Determine the basis functions ψ 1 (t) and ψ 2 (t) in the bandpass range. (b) Draw the real and the imaginary part of u tx,l (t) for the following data sequences d 1,...,d 5 = +1,+1, 1,+1, 1,+1, 1 d 1,...,d 5 = 1, 1,+1, 1,+1,+1, 1 (c) Determine u tx,l (t). (d) Draw the phase θ(t) of u tx,l (t) in the interval t < 5. (e) Derive the instantaneous frequency f i (t) = 1 range for t < 5. 2π d θ(t) d t in the equivalent baseband 11

12 4.13 he elementary signals of 2-FSK with a frequency separation f = 1/ are { U0 cos (2πf s 1 (t) = c t) for 0 t < and 0 else { U0 cos ( 2π ( f s 2 (t) = c + 1 ) ) t for 0 t < 0 else where f c = k/, k 1, integer. he transmitted signal can be also written as u tx (t) = U 0 2 rect where d n { 1,+1}. [ ( ( (d n + 1)cos (2πf c t) (d n 1)cos 2π f c + 1 ) )] t ( ) t /2 n, (1) (a) Determine the transmit signal in the equivalent baseband, if the transformation will be done with respect to the frequency f c + 1/(2). (b) Derive the power spectral density., 12

13 5 Discrete signals and systems 5.1 Consider the following time discrete filter. u 1 (t) t 0 t 0 t 0 u 2 (t) (a) Determine the impulse response g(t) and the transfer function G(f) of the filter. he input signal u 1 (t) is given as 2 u 1 (t) = c 1 [n]δ(t n t 0 ), n=0 with c 1 [0] = 1 c 1 [1] = 2 c 1 [2] = 3. (b) Determine the output signal u 2 (t) based on a matrix-vector notation. Distinguish the two possibilities. (c) Is the system causal and time-invariant? Justify your answer also on the basis of the system matrix C. (d) Modify the relation between u 2 (t) and u 1 (t) such, that the system matrix becomes cyclic. he cyclic matrix is denotes as C circ. (e) Specify the Eigen-vectors of the cyclic system matrix. 13

14 6 Additional asks 6.1 A periodic train of rectangular pulses can be considered as a special case of a cyclostationary random process, if it is assumed, that all realization exhibit the same synchronization. In the following, the discrete time signal u p [n], n G shall be considered, where the elements can be defined using the following vector notation u p = [ u p [0] u p [1]... u p [6] u p [7] ] = [ ]. (a) Determine the second moment E{u 2 p[n]} (Ensemble-averaging)). (b) Compare the result with the time average A{u 2 p[n]} = n=0 u2 p[n]. (c) Determine the discrete ACF ϕ pp [n,n + m] = E{u p [n]u p [n + m]}, n,m G, and show, that ϕ pp [n,n + m] is cyclo-stationary. (d) Determine the time averaged ACF A{ϕ pp [n,n + m]} = ϕ pp [n,n + m]. (e) Compare the result with the ACF ϕ pp [m] = n=0 u p[n]u p [n + m], which represents the definition of the ACF for periodic signals. (f) What would happen with respect to the ACF E{u p [n]u p [n+m]}, if the realizations would exhibit random delays between 0 and 7? 6.2 he electrical field (considered in the equivalent lowpass range) at the output of a laser diode can be written as e(t) = E 0 e +jφ(t) where φ(t) represents phase noise. he phase noise determines the power spectral density of e(t). he phase noise is assumed to be zero mean Gaussian distributed. It is assumed that the instantaneous frequency f i can be modeled as a WSS random process with the power spectral density N 0 (technical). (a) What is the expression for the autocorrelation function ϕ EE (τ) of the electrical field e(t)? (b) Determine the power spectral density Φ EE (f) of e(t), if no phase noise would occur (ideal laser). (c) Rewrite the autocorrelation function (part (a)) by using the following expression n=0 E{e jx } = e 1 2 E{X2}, which is valid for a zero mean Gaussian distributed random variable X. (d) Determine the autocorrelation function ϕ (λ) of the phase difference (t) = φ(t + τ) φ(t). (e) Is the phase noise a stationary process? Determine the mean power of the phase noise. 14

15 (f) Determine the autocorrelation function ϕ EE (τ) and the PSD Φ EE (f). Use the assignment N 0 = 2f h π. 6.3 A dispersive channel with impulse response g(t) is assumed, where g(t) = 0 for t < 0 and t > τ max. (a) Draw an exemplary impulse response g(t). (b) Calculate the received signal u 2 (t) for the input signal u 1 (t) = U 0. (c) Calculate the received signal u 2 (t) for the input signal u 1 (t) = U 0 e j2πf 1t, where f 1 > 0, real. (d) Calculate the received signal u 2 (t) for t τ max, if the input signal is given as u 1 (t) = s(t) U 0, where s(t) denotes the step function. (e) Calculate the received signal u 2 (t) for t τ max, if the input signal is given as u 1 (t) = s(t) U 0 e j2πf 1t. (f) Calculate the received signal ( u 2 (t) for t [τ max, + τ max ], if the input signal is given as u 1 (t) = U 0 rect t (+τmax)/2 +τ max ). 6.4 A dispersive channel with impulse response g(t) is assumed, where g(t) = 0 for t < 0 and t > τ max. (a) Calculate the received signal u 2 (t) for the input signal ( ) t /2 u 1 (t) = U 0 rect. It is advantageously to analyze u 2 (t) for different time intervals separately. (b) Calculate within the interval [0,]? u 3 (t) = u 2 (t) + u 2 (t + ) 15