Communication Systems Lecture 1, Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University 1
Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf
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
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
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
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
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
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
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
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
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
Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 1
Noise A random unwanted signal We will look at two types white noise: Gaussian white noise colored noise Effect of filtering noise 13
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
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
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
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
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
Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 19
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
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
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 ( )
Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 3
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
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
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
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
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
Narrowband Noise Alternative format (Proakis pp.197-198) ( ) = ( ) = 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
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
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
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
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
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
Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise Noise equivalent bandwidth Narrowband noise Ricean pdf 35
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
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
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
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
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
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