Communication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University

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1 Communication Systems Lecture 1, Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University 1

2 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf

3 Linear Systems From Chapter x(t) H y(t) Linear systems: [ ] y ( t) = H a x ( t) + a x ( t) 1 1 = a1h [ x1( t) ] + ah [ x( t) ] Time invariant systems: [ ] H [ ] y ( t) = H x( t) y ( t t ) = x( t t ) 0 0 Linear time invariant (LTI) systems: Both linear and time invariant 3

4 Linear Systems x(t) h(t) y(t) Impulse response of an LTI system h( t) = H [ δ ( t) ] Any input can be written in terms of δ ( t ) x() t = x( λ) δ( t λ) dλ Send x(t) to an LTI system h(t): yt ( ) = H [ xt ( )] = H x( λ ) δ( t λ) dλ = x( λ) ht ( λ) dλ = xt ( λ) h( λ) dλ In freq domain: Y ( f ) = H ( f ) X ( f ) Only need to know h(t) to get y(t) 4

5 Input-Output Relationship for Linear Systems with WSS Inputs Energy signals (with finite energy): E x ( ) d 1 ( ) ( ) X d = τ τ = ω ω= π X f df Energy spectral density: G( f) X f ( ) x(t) h(t) y(t) Y ( f ) = H ( f ) X ( f ) y * ( ) = ( ) ( ) = H ( f) X( f) = H( f) G ( f) G f Y f Y f x 5

6 Input-Output Relationship for Linear Systems with WSS Inputs Power signals have finite power, but infinite energy. R( τ) E[ x() t x( t+ τ) ] Power spectral density of WSS processes: After filtering by linear system: x(t) jπfτ S( f ) F R( ) = R( ) e d h(t) Y ( f ) = H ( f ) X ( f ) y(t) τ τ τ (assume all signals are real) y ( ) = ( ) ( ) S f H f S f x 6

7 Input-Output Relationship for Linear Systems with WSS Inputs x(t) Proof: Ryy ( τ) = Eytyt [ () ( + τ) ] = Eyt () huxt ( ) ( + τ udu ) ( ) [ τ ] ( ) = hueytxt () ( + u) du= hur ( τ udu ) = h( τ) R ( τ) yx h(t) y(t) S ( f) = H( f) S ( f) ( ) ( ) ( ) yt= huxt udu y yx x ( ) = ( ) ( ) In freq domain: S yy f H f S yx f 7

8 Input-Output Relationship for Linear Systems with WSS Inputs [ ] [ ] R ( τ) = E y() t x( t+ τ) = E x() t y( t τ) = R ( τ) yx In freq domain: S f S f * yx ( ) = ( ) xy xy Rxy ( τ) = Extyt [ () ( + τ) ] = Ext () huxt ( ) ( + τ udu ) ( ) = hur ( τ udu ) = h( τ) R ( τ) xx In freq domain: xy xx ( ) = ( ) ( ) S f H f S f xx 8

9 Input-Output Relationship for Linear Systems with WSS Inputs R ( τ) = R ( τ) = h( τ) R ( τ) = h( τ) R ( τ) yx xy xx xx R ( τ) = h( τ) R ( τ) = h( τ) h( τ) R ( τ) yy yx xx In freq domain: y ( ) = ( ) ( ) S f H f S f x 9

10 Example Multipath X( t ) Y( t) Σ T ( ω) 1 H j = e ω j T output PSD jωt + jωt ( ω) = ( 1 )( 1 ) H j e e ( ωt) = 1 cos ωt + 1= 1 cos 10

11 cont we thus have Y ( ω) = ( ω)( 1 cosω ) S S T X we can easily see why some channels get wiped out ω/tt 11

12 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 1

13 Noise A random unwanted signal We will look at two types white noise: Gaussian white noise colored noise Effect of filtering noise 13

14 White Noise White noise has constant power spectral density: 1 S( f) = N0, - <f<. 1 N0 : double-sided power spectral density. N : single-sided power spectral density. 0 The total power is infinite: Different samples of white noise are uncorrelated: 1 R( τ) = N0δ( τ). S( f) df =. R(0) : infinite. 14

15 Gaussian Noise Noise whose amplitude has Gaussian pdf. This says nothing of the correlation of the noise in time or of the spectral density of the noise: Gaussian noise and white noise are two different concepts. Neither implies the other. 15

16 Gaussian White Noise Noise with a constant power spectral density and a Gaussian distribution of amplitude. Gaussian white noise is a good approximation of many real-world situations and generates mathematically tractable models. Samples of Gaussian white noise are independent: Uncorrelated and independent are same for Gaussian pdf. 16

17 Filtered White Noise A white noise with psd 1/ N0 is filtered by h(t): xy ( ) = ( ) ( ) 0 ( ) ( ) S f H f S f xx S f = H f 1 1 Rxy ( τ) = h( τ) Rxx ( τ) = h( τ) N0δ( τ) = N0h( τ) xy N 17

18 Filtered White Noise ( ) Example N t Y( t) N(t): white noise with psd 1/ N 0. S YY ( f) ( ) ht H(f) 1 B No / f B = 0 otherwise B 1 No jπfτ RYY ( τ) = F SYY ( f) = e df B No 1 N 1 = e = e e jπτ jπτ ( ) jπfτ B o jπbτ jπbτ B No 1 sinπbτ = sin πbτ = BNo = BNosinc B πτ πbτ ( τ) f 18

19 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 19

20 Noise Equivalent Bandwidth N( t) ht ( ) Y( t) 1 Nt ( ) : white noise with psd N0. Find the bandwidth of an ideal filter that has the same midband gain as H(f) and passes the same noise power 0

21 Noise Equivalent Bandwidth Output power: P = S ( f ) df = H ( f ) S ( f ) df = N H ( f ) df 1 yy xx 0 Output power of an ideal filter with gain H 0 and single-side bandwidth B: 0 Let P 1 = P : 1 P ( ) = N0 H0B = N0H0B 1 B = H( f) df H 0 0 1

22 Noise Equivalent Bandwidth Time domain expression: Assumption: low-pass filter with max gain at f = 0. N 0 N 0 H ( f) df = H0 B Rayleigh's energy theorem: H ( f) df = h( t) dt jπ ft 0 f= 0 f= 0 H = H( f) = h( t) e dt = h( t) dt B ht () dt = htdt ( )

23 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 3

24 Narrowband Noise Narrowband bandpass noise is common receivers have RF and IF filters Recall that for bandpass deterministic signals () = () + () xt x t jx t R -fc x ( t) = x ( t) e j π fct p I X(f) fc+b/ f fc xt = Re x( t) = x tcos ω t x tsin ω t ( ) { } ( ) ( ) ( ) ( ) p R c I c 4

25 Definition of Bandpass or Narrowband Random Process (Proakis pp. 195) Definition: A random process X(t) is bandpass or narrowband random process if its power spectral density S X (f) is nonzero only in a small neighborhood of some high frequency f 0. The definition of bandpass random process is the generalization of bandpass deterministic signals. Deterministic signals: defined by its Fourier transform Random processes: defined by its power spectral density, because R ( τ) is deterministic. X Note: f 0 needs not be the center of the signal bandwidth, or in the signal bandwidth at all (see Example 5.1) S X (f) -fo fo fc+b/ f 5

26 Narrowband Noise Narrowband noise n(t): random process Quadrature component representation: ( ) = ( ) cos( ω + θ) ( ) sin( ω + θ) nt n t t n t t c c s c θ : arbitrary phase angle. n c (t): LP in-phase component (random process) n s (t): LP quadrature component (random process) Envelope representation: ( ) = ( ) cos ( ω + φ( ) + θ) nt Rt t t Rt n t n t () = c() + s() c 1 ns () t φ() t = tan n () t c 6

27 Narrowband Noise Generating n c (t) and n s (t) from n(t): Ziemer This is a linear system with respect to the input: If n(t) is Gaussian, n c (t) and n s (t) will be Gaussian. 7

28 Narrowband Noise Some properties: Mean Variance ( ) = ( ) = ( ) = 0. Ent En t En t c s En t En t En t N. Power spectral density: ( ) = ( ) = ( ) c s ( ) = ( ) = Lp ( ) + ( + ) S f S f S f f S f f c s n n n c n c ( ) = Lp ( ) ( + ) S f j S f f S f f c s nn n c n c LP[ ]: Lowpass result after filtering by H(f). 8

29 Narrowband Noise Alternative format (Proakis pp ) ( ) = ( ) = Lp ( ) + ( + ) S f S f c s S f f S f f Sn( f fc) + Sn( f + fc), f f0, = 0, otherwise n n n c n c ( ) = Lp ( ) ( + ) Snn f j S c s n f fc Sn f f c js n( f fc) Sn( f+ fc), f f0, = 0, otherwise 9

30 1. Proakis derivation is based on Hilbert transform, No need to use Fig. 5.1, which is confusing (which is know and which is unknown) n( t) = nt ()cos ( πft) + nt ˆ()sin( πft) Xˆ c s ( ) = ˆ( )cos ( π ) ()sin( π ) if S x (f) has no DC component. 0 0 n t nt ft nt ft XXˆ 0 0 R ( τ) = R ( τ), R ( τ) = R ( τ) (Proakis pp. 176) X Autocorrelation and psd are all derived from this. Note the R xx (t) definition of ours is diff from Proakis X 30

31 Narrowband Noise PSD Prove also independent of n(t) () = () ( ω + θ) z1 t n t cos ct () = () cos( ω + θ) () sin( ω + θ) nt n t t n t t ( τ) = 4 ( ) ( + τ) cos( ω + θ) cos( ω + ω τ+ θ) R E n t n t t t 1 z c c c = Entnt () ( + τ) cosωcτ+ cos( ωct+ ωcτ+ θ) = Entnt () ( + τ) E[ cosωcτ] + E cos( ωct+ ωcτ+ θ) = R τ cosω τ n ( ) c c c s c ( ) = ( ) = Lp ( ) + ( + ) S f S f S f f S f f c s n n n c n c Assume that θ is uniform on [0, π] From the circuit: 1 ( ) = ( ) + ( + ) S f S f f S f f Z n c n c 31

32 cont After Lowpass filtering by H(f) ( ) = ( ) + ( + ) S f S f f S f f Z n c n c 1 ( ) = Lp ( ) + ( + ) S f S f f S f f c n n c n c N 0 / Sn(f) Sn(f-fc) fc fc+b/ f f Sn(f+fc) Snc(f) f f σ = σ = σ n n n c (integrate over all f) s 3

33 Cont S ( f) = jlp S ( f f ) S ( f + f ) nn n c n c c s Sn(f) N 0 / f fc fc+b/ Sn(f-fc) f Sn(f+fc) Sn(f-fc)-Sn(f+fc) = 0 if symm If S n (f) is symmetric around fc: n c (t) and n s (t) are uncorrelated. nn c s f ( ) 0 S f = If n(t) is Gaussian (S n (f) sym), n c (t) and n s (t) are independent. 33

34 Cont If S n (f) is symmetric around f c, and if n(t) is Gaussian, n c (t) and n s (t) are independent. f n,; t n, t ( ) c s + τ = Envelope representation: then 1 πn e c+ ns n N Rt () = n() t + n(), t () t = tan f r ( ) f r () 1 ns () t c s φ nc () t r πn r N, φ =, r>0, φ π. = r N e r e N, r>0: Rayleigh pdf. 34

35 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 35

36 Ricean PDF Consider sum of a random phased sinusoid and bandlimited Gaussian random noise: () = cos( ω + θ) + () cos( ω ) () sin( ω ) z t A t n t t n t t 0 c 0 s 0 Ricean pdf: the pdf of the envelope of this stationary process: r + A r Ar fr r e I 0 σ σ Modified Bessel function of order 0: σ () =, r 0. 1 π I0 ( u) = exp( ucosα ) dα π 0 36

37 Ricean PDF Derivation: ( ) = cos( ω + θ) + ( ) cos( ω ) ( ) sin( ω ) zt A t n t t n t t 0 c 0 s 0 [ cosω cosθ sin ω sin θ] () cos( ω ) () sin( ω ) = A t t + n t t n t t 0 0 c 0 s 0 () () = Acosθ nc t cosω0t Asin θ ns t + + sin ω0t = X()cos t ω t Y()sin t ω t 0 0 ( ) X() t = Acosθ + n t ( ) Yt () = Asinθ + n t Objective: find pdf of R(t). s c R t X t Y t () = () + () () t tan Yt () Xt () 1 Φ = 37

38 Ricean PDF ( ) X() t = Acosθ + n t ( ) Yt () = Asinθ + n t f = XY ( x, y θ ) = s c n(t) Gaussian nc(t), ns(t) independent Gaussian. ( ( x A ) ) ( x A ) ( ( ( ) ) x + y A x θ + y θ + A σ ) ( ) exp cos θ / σ exp sin θ / σ πσ πσ exp cos sin / πσ Let x = r cosφ ( xy, ) frφ ( r, φ θ) = fxy( x( r, φ), y( r, φ) θ). y= rsin φ, (, r φ) r frφ (, r φ θ) = exp r + A Arcos θ φ /σ πσ ( ( ( )) ) 38

39 Ricean PDF r frφ (, r φ θ) = exp r + A Arcos θ φ /σ πσ Integrating over φ ( ( ( )) ) r r + A 1 π Ar fr () r = exp exp cos ( θ φ) dφ σ σ π 0 σ r r + A Ar = exp I 0 σ σ σ 1 π I0 ( u) = exp( ucosα ) dα π 0 Modified Bessel function of order 0: I 0 ( 0) = 1 r r When A=0, fr ( r) = exp : σ σ Rayleigh pdf. 39

40 Ricean PDF ( ) = cos( ω + θ) + ( ) cos( ω ) ( ) sin( ω ) z t A t n t t n t t Define Ricean K-factor K = : Signal to noise power ratio 0 c 0 s 0 A σ r r + A Ar fr () r = exp I0 σ σ σ r r r fr () r = exp K I 0 K σ σ σ 40

41 Ricean PDF Rt X t Y t () = () + () ( θ c() ) θ s() ( ) E R () t = E X () t + E Y () t = E Acos + n t + E Asin + n t ( cos θ sin θ) () () c s cosθ c() sinθ s() = E A + A + E n t + E n t + AE n t + n t σ σ 1 ( K) = + = + A 41