ECE 461 Fall 2006September 12, 2006Signal Space ConceptsIn order to proceed with th e design and analysis of digital communication systems (in complexbaseband) it is important for us to understand some properties of the s pace in which the complexmessage bearing signal s(t) lies.Inner Product and Norm• Let x(t) and y(t) be complex valued signals with t ∈ [a, b]. If a and b are not s pecified, it isassumed that t ∈ (−∞, ∞).Definition 1. (Inner Product)< x(t), y(t) >∆=Zbax(u)y∗(u)du . (1)The inner product satisfies the necessary axioms:➀ < x(t), y(t) >=< y(t), x(t) >∗➁ < x(t) + y(t), z(t) >=< x(t), z(t) > + < y(t), z(t) >➂ < αx(t), y(t) >= α < x(t), y(t) >➃ < x(t), x(t) >≥ 0, and < x(t), x(t) >= 0 iff x(t) = 0 for all t.• Signals x(t) and y(t) are said to be orthogonal if < x(t), y(t) >= 0. The orthogonality of x(t) andy(t) is sometimes denoted by x(t) ⊥ y(t).Definition 2. (Norm) The inn er product defi ned above induces the following norm:kx(t)k =p< x(t), x(t) > . (2)It is easy to show that the above quantity is a valid norm in th at it satisfies the required axioms.(Based on ➃ above, all that one needs to verify is the triangle inequality.)Prop e rties of Inner Product and Norm➀ Cauchy-Schwarz Inequality:| < x(t), y(t) > | ≤ kx(t)kky(t)k (3)with equality iff x(t) = αy(t) for some complex α.➁ Parallelogram Law:kx(t) + y(t)k2+ kx(t) − y(t)k2= 2kx(t)k2+ 2ky(t)k2. (4)cV.V. Veeravalli, 2006 1➂ Pythagorean Theorm: If x(t) ⊥ y(t) thenkx(t) + y(t)k2= kx(t)k2+ ky(t)k2. (5)Signal Space and Basis Functions• If all we know about the signal s(t) is that it h as finite energy, i.e., ks(t)k < ∞, then we canconsider s(t) to belong to the (infinite-dimensional) Hilbert space of complex signals with finiteenergy and with inner product as d efi ned above. This Hilbert space is denoted by L2[a, b].One can find a (countably infinite) set of functions {fi(t)}∞i=1in L2[a, b] that are orthornormal,i.e.,< fi(t), fℓ(t) >= δiℓ, such that for any s(t) ∈ L2[a, b], we haves(t) =∞Xi=1sifi(t) . (6)The set {fi(t)}∞i=1is called a complete basis for L2[a, b]Example 1. On L2[0, T ], we have the Fourier basis, defined by:fi(t) =1√Tej2πit/T, i = 0, ±1, ±2, . . . (7)• Suppose we further impose constraint that the complex baseband signal s(t) is approximatelybandlimited to W/2 Hz (and time-limited to [−T/2, T/2], say), and impose no other constraintson the s ignal space. Then the appropriate basis functions for the signal space are the ProlateSpheroidal Wave Functions (PSWF’s). See the p apers by Slepian, Landau and Pollack for adescription of PS WF’s. This basis is optimum in the sense that, although there are a countablyinfinite number of functions in the set, at most W T of these are enough to capture most ofthe energy for any signal in this signal space. So the signal space of complex signals that areapproximately bandlimited to W/2 Hz and time limited to [−T /2, T/2] is approximately finitedimensional.• More typically in communication systems, s(t) is one of M possible signals s1(t), s2(t), . . . ,sM(t). If we let S = span{s1(t), . . . , sM(t)}, then dim(S) = n ≤ M. The signal s(t) can thenbe considered to belong to the n -dim space S. One can find an orthonormal basis for S by thestandard Gram-Schmidt procedure:g1(t) = s1(t), f1(u) =(g1(u)kg1(t)kif kg1(t)k 6= 0stop otherwise(8)g2(u) = s2(u)− < s2(t), f1(t) > f1(u), f2(u) =(g2(u)kg2(t)kif kg2(t)k 6= 0stop otherwise(9)gℓ(u) = sℓ(u) −ℓ−1Xi=1< sℓ(t), fi(t) > fi(u), fℓ(u) =(gℓ(u)kgℓ(t)kif kgℓ(t)k 6= 0stop otherwise(10)cV.V. Veeravalli, 2006 2• For signal s(t) ∈ S, we can writes(t) =nXℓ=1sℓfℓ(t) , with sℓ=< s(t), fℓ(t) > . (11)The signal s(t) ∈ S is equivalent to the vector s = [s1s2···sn]⊤in the sense thatks(t)k =√s†s = ksk (show this!) (12)and for sk(t), sm(t) ∈ S< sk(t), sm(t) >= s†msk=< sk, sm> (show this!) . (13)Signal Energy, Correlation and Distance• The energy of a signal s(t) is denoted by E and is given byE = ks(t)k2. (14)• The correlation between two signals sk(t) and sm(t), which is a measure of the similarity betweenthese two signals, is given byρkm=< sk(t), sm(t) >ksk(t)kksm(t)k=< sk(t), sm(t) >√EkEm. (15)• The distance between two signals sk(t) and sm(t), which is also a measure of the similarity betweenthese two signals, is given bydkm= ksk(t) − sm(t)k =Ek+ Em− 2pEkEmRe[ρk,m]12. (16)If Ek= Em= E, thendk,m= [2E(1 −Re[ρk,m])]12. (17)cV.V. Veeravalli, 2006