TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem. The "random walk" was modelled as a random sequence [ n] where W[i] are binary i.i.d. random variables with P[W[i] = s] = p (orward step with probability p) P[W[i] = s] = p (backward step with probability p) n i W[ i] To illustrate, one possible sample path (realization) could look something like 3s s s -s -s [n] ut keep in mind that the above igure is just an example (one realization). We are really talking about statistical characterizations here. n TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall In order to the sequence to be wide-sense stationary (WSS), the autocovariance (or autocorrelation) unction should be shit invariant and the mean unction should be constant (see Carlson 4 th Ed. p. 357-359). This means also that the variance K (m,m) o the sequence should be constant in time as well. Now K but K ( m, m) ( m l, m l) ( m) m(s) p( p) ( m l) ( m l)(s) p( p) ( m) meaning that the variance o this random sequence is increasing in time. Thus, the process is not WSS. Furthermore, the mean unction is ( n) ns( p ) which is also time-dependent (in general) implying non-stationarity. Thus, we conclude that the sequence is NOT WSS! This signal model can also be easily examined using Matlab. In the ollowing, we generate independent realizations (o length samples) with s = and p = /. So according to the theory, the mean and variance are in this case given by [ n] E [ n] ns(p) n ( n) E [ n] [ n] E [ n] min( n, n)() (/4) n These results can now be veriied by calculating the corresponding ensemble averages over the generated independent realizations. The results are illustrated in general in the ollowing two igures. / 6 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Three Independent Realizations 4 4 6 5 5 4 5 5 4 6 8 5 5 Time Index 5 Ensemble Mean and Variance (Estimates) TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem. Oscillators are ubiquitous in physical systems, especially electronic and optical ones. For example, in radio requency communication systems they are used or requency translation o inormation signals and or channel selection. Oscillators are also present in digital electronic systems which require a time reerence, i.e., a clock signal, in order to synchronize operations. Noise is o major concern in oscillators, because introducing even small noise into an oscillator leads to dramatic changes in its requency spectrum and timing properties. This phenomenon is known as phase noise or timing jitter. A perect oscillator would have localized tones at discrete requencies (i.e., harmonics), but any corrupting noise spreads these perect tones, resulting in high power levels at neighbouring requencies. This eect is the major contributor to undesired phenomena such as interchannel intererence, leading to increased bit error rates (ER s) in RF communication systems. Another maniestation o the same phenomenon, jitter, is important in clocked and sampled data systems. Uncertainties in switching instants caused by noise lead to synchronization problems. Characterizing how noise aects oscillators is thereore crucial or practical applications. 5 variance In our case, we study the eects o a small phase noise present in a j( t( t)) complex oscillator signal, xt () Ae. In ideal case, () t, and we have only a discrete impulse at requency in the spectral density unction G (). Let s start by rewriting the given signal x(t) as j( t( t)) j t j () t j t x() t Ae Ae e Ae v() t. 5 mean It is important to note that v(t) is a random process and thereore x(t) is also a random process! First we study is the x(t) a WSS process and we start by taking the expected value (or ensemble average) with t held ixed at arbitrary value, as 5 5 5 Time Index 3 / 6 4 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall x t E x t E Ae v t Ae E[()] v t () [ ()] [ j t ()] j t ased on the small noise phase assumption ( () t rad), we can approximate v(t) as vt e cos t jsin t j ( t), j () t ( ) ( ( )) ( ( )) and the expected value simpliies to x t Ae E j t Ae. () j t [ ()] j t We notice that the expected value depends on time, and thereore x(t) is not a WSS process! Even though, x(t) is not a WSS process, we can still calculate the autocorrelation o the random process. When we investigate the relationship between two random variables, we use autocorrelation unction which is deined or complex random unctions as R ( t, t ) E[ x( t ) x*( t )]. This unction measures the relatedness or dependence between two random processes. Now that we have the required tools, let s start with the derivation. R ( t, t ) E[ x( t ) x*( t )] [ ( ) *( )] j t j( t) E Ae v t Ae v t j ( tt) Ae Evt [()*( v t)]. TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Recalling the small noise phase assumption ( () t rad), the expected value E[v(t )v*(t )] simpliies to E[ v( t) v*( t)] E[( j ( t))( j ( t))] E[ ( t ) ( t )]. E[ ( t)] I we now assume that the random variable () t is WSS, we are only interested in the time dierence between the unctions and thereore we mark t =t and t =t-τ. Thus, the autocorrelation unction o x(t) can be rewritten as j ( ) { E[ ( ) ( )]} j R A e t t Ae ( R( ) ) j j Ae Ae R(). We notice that the dependency orm time t disappears and that the autocorrelation unction depends only in the time dierence τ. Now, in the autocorrelation unction we have the original spectral impulse at and in addition we have requency shited version o the autocorrelation unction o () t around requency. Now it is easy to deine the Fourier transorm o R x (τ), which represents the spectral density unction G x (). G( ) F{ R ( )} A ( ) A G ( ). ecause we assumed that () t is a realization rom WSS process, there exists a Fourier transorm pair R ( ) G ( ). 5 / 6 6 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall In the spectral density unction it is even clearer that the power spectrum o the phase noise is shited around requency. The idea is even urther illustrated below. This holds when the phase noise is small. TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem 3. The variance o the thermal noise voltage v(t) at the open-circuit terminals o a resistor R at temperature T is given by (Carlson 4 th Ed. p.37) G ( ) Gx ( ) vt () kt v R where 3h k = oltzmann constant =.38* -3 h = Planc constant = 6.6* -34 In the book/course notes, the spectral density unction or the noise process is given as G ( ) x Rh G V ( ) h / kt e So, we immediately note that the process is not exactly white (uncorrelated) because the spectral density unction varies with requency. We could calculate the autocorrelation unction or the process by inverse Fourier transorm o G V (). This autocorrelation unction at delay tells us the value o the covariance or t i - t j =, since the mean is zero (when the mean is zero, autocovariance = autocorrelation). We can calculate the total power o the process directly with the help o the irst ormula or v or by integrating G V () rom -ininity to ininity. We do that with Maple and get 7 / 6 8 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall > k:=.37*^(-3);r:=;t:=9;h:=6.6*^(-34); - k :=.37 R := T := 9-33 h :=.66 > G:=*R*h*abs()/(exp(h*abs()/(k*T))-): > total_power_:=eval(*(pi*k*t)^*r/(3*h)); -5 total_power_ :=.5688746 > total_power_:=int(g,=-ininity...ininity); -5 total_power_ :=.5688746 To get the desired approximation or G V () we proceed as ollows. First we can write G V () as Rh ( ) h GV Rh h / kt e kt h! kt using the terms up to nd order o the series expansion or e x o the orm e x x! x... TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Using only the terms up to the nd order is justiied under the given assumption that h kt. The given orm can be simpliied to G V ( ) h Rh kt h! kt h RkT kt h kt Rh h kt Then we use the trick that (+x) - -x when x is small and we get (use x = h /kt) the desired result h G V ( ) RkT kt The approximation and true density are given in the ollowing Figure. The power in some requency range is given by the integral o G V () over the desired requencies. The integration is easy using the approximation and can be carried out by pen and paper. For example or = - P G ( ) d V symmetry h G V ( ) d 4RkT 4kT Remember, however, that the result is valid only i kt/h. 9 / 6 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall 9 x 9 8. x 6 True Approximation Spectral Density 7 6 5 4 3 Power.8.6.4 True Approximation 6 7 8 9 Frequency [Hz] Figure : Approximative and true power spectral densities (notice the logarithmic scaling o the requency axis).. Maximum Frequency (andwidth) [Hz] Figure : The power in the requency range < using both approximative and true densities. We can also calculate the wanted noise power in the requency range < GHz with Maple, using both the accurate and approximative densities: > G:=*R*k*T*(-h*abs()/(*k*T)): > power_:=int(g,=-^exponent...^exponent): > power_:=int(g,=-^exponent...^exponent): > eval(subs(exponent=9,power_)); -8.588537999 > eval(subs(exponent=9,power_)); -8.588538 Clearly, the approximation is good. In general, the power in the requency range is illustrated in the ollowing Figure using both the approximative and true densities. Finally it was asked how big a portion o the total power is within the requency range < GHz, so >raction:=*eval(subs(exponent=9,power_))/total_power_; raction :=.53485 meaning that only.% o the total noise power is within the requency range < GHz - so 99.9% o the total noise power is outside that range at higher requencies ( GHz and above)! o sounds somewhat strange (?) but this is indeed the case!! o this simply relects the act that we are really dealing with an extremely wideband (close to white) noise process here!! / 6 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem 4. When white noise with spectral density / is iltered with H(), the spectral density o the ilter output signal is / H(). This being the case, the power N o the output signal is given by N H ( ) d H ( ) d Noise equivalent bandwidth N is now deined as (see Carlson 4 th Ed. p.378) N g H ( ) d where g = max H(). This means that N is the bandwidth o the ideal ilter having the same maximum power gain and output power as H(). utterworth ilters are characterized with (n = ilter order and = 3d bandwidth) H ( ) / n meaning that g = or all utterworth ilters. We can then use the deinition to get N g H ( ) d / n d TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall To choose the easy way out, we can check some table o standard integrals (e.g. Carlson 4 th Ed., p. 786) and note that / k dx sin( / k) x k, k > Now, use the substitution x=/ and dx=d/. N nsin( / n) sin( / n) / n d n / x and in the limit we get lim lim n sin( / n) / n N n n / n dx sin( / n) because sin(x)/x as x (Hint: Prove with L Hospitals rule). Results are easy to check with Maple: > assume(k>);int(/(+x^k),x=..ininity); > N:=/(sin(Pi/(*n))/(Pi/(*n))); Pi ------------ Pi sin(----) k k Pi N := / --------------- Pi sin(/ ----) n n > limit(n,n=ininity); 3 / 6 4 / 6
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Noise Equivalent andwidth N 9 8 7 6 5 4 3 4 6 8 4 6 8 Filter Order n Figure 3: Noise equivalent bandwidth N as a unction o the ilter order n or a utterworth ilter with a 3d bandwidth o Hz. H().9.8.7.6.5.4.3.. n = n = 4 n = 3 4 5 Frequency [Hz] Figure 4: Squared amplitude responses or three utterworth ilters with 3d bandwidth = Hz..9 3 d bandwidth = Hz In this case the, noise equivalent bandwidth approaches the 3d bandwidth quite rapidly when the ilter order is increased. This means that the utterworth ilters become closer and closer to "ideal" ilters in the average output power sense when the ilter order is increased. With some other ilter structures, this behaviour can be dierent (slower/aster). H().8.7.6.5.4 Noise equivalent bandwidth The ollowing igures illustrate the concept o noise equivalent bandwidth urther.3.. 3 4 5 Frequency [Hz] Figure 5: Squared amplitude response or a utterworth ilter o order n = with 3 d bandwidth = Hz, the noise equivalent bandwidth is also in the igure. 5 / 6 6 / 6