ECE 461 Fall 2006

September 12, 2006

Signal Space Concepts

In order to proceed with the design and analysis of digital communication systems (in complex
baseband) it is important for us to understand some properties of the space in which the complex
message 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 specified, it is
assumed that t ∈ (−∞, ∞).
Definition 1. (Inner Product)
Z b
∆
< x(t), y(t) > = x(u)y ∗ (u)du . (1)
a

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) and
y(t) is sometimes denoted by x(t) ⊥ y(t).
Definition 2. (Norm) The inner product defined above induces the following norm:
p
kx(t)k = < x(t), x(t) > . (2)

It is easy to show that the above quantity is a valid norm in that it satisfies the required axioms.
(Based on ➃ above, all that one needs to verify is the triangle inequality.)

Properties of Inner Product and Norm

➀ 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)

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➂ Pythagorean Theorm: If x(t) ⊥ y(t) then

kx(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 has finite energy, i.e., ks(t)k < ∞, then we can
consider s(t) to belong to the (infinite-dimensional) Hilbert space of complex signals with finite
energy and with inner product as defined above. This Hilbert space is denoted by L2 [a, b].
One can find a (countably infinite) set of functions {fi (t)}∞ i=1 in L2 [a, b] that are orthornormal,
i.e.,
< fi (t), fℓ (t) >= δiℓ , such that for any s(t) ∈ L2 [a, b], we have
∞
X
s(t) = si fi (t) . (6)
i=1

The set {fi (t)}∞

i=1 is called a complete basis for L2 [a, b]
Example 1. On L2 [0, T ], we have the Fourier basis, defined by:
1
fi (t) = √ ej2πit/T , i = 0, ±1, ±2, . . . (7)
T

• Suppose we further impose constraint that the complex baseband signal s(t) is approximately
bandlimited to W/2 Hz (and time-limited to [−T /2, T /2], say), and impose no other constraints
on the signal space. Then the appropriate basis functions for the signal space are the Prolate
Spheroidal Wave Functions (PSWF’s). See the papers by Slepian, Landau and Pollack for a
description of PSWF’s. This basis is optimum in the sense that, although there are a countably
infinite number of functions in the set, at most W T of these are enough to capture most of
the energy for any signal in this signal space. So the signal space of complex signals that are
approximately bandlimited to W/2 Hz and time limited to [−T /2, T /2] is approximately finite
dimensional.

• 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 then
be considered to belong to the n-dim space S. One can find an orthonormal basis for S by the
standard Gram-Schmidt procedure:
( g (u)
1
if kg1 (t)k =
6 0
g1 (t) = s1 (t), f1 (u) = kg1 (t)k (8)
stop otherwise
( g (u)
2
if kg2 (t)k =
6 0
g2 (u) = s2 (u)− < s2 (t), f1 (t) > f1 (u), f2 (u) = kg2 (t)k (9)
stop otherwise
ℓ−1
( g (u)
X ℓ
if kgℓ (t)k =
6 0
gℓ (u) = sℓ (u) − < sℓ (t), fi (t) > fi (u), fℓ (u) = kgℓ (t)k (10)
i=1 stop otherwise

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• For signal s(t) ∈ S, we can write
n
X
s(t) = sℓ fℓ (t) , with sℓ =< s(t), fℓ (t) > . (11)
ℓ=1

The signal s(t) ∈ S is equivalent to the vector s = [s1 s2 · · · sn ]⊤ in the sense that
√
ks(t)k = s† s = ksk (show this!) (12)

and for sk (t), sm (t) ∈ S

< sk (t), sm (t) >= s†m sk =< sk , sm > (show this!) . (13)

Signal Energy, Correlation and Distance

• The energy of a signal s(t) is denoted by E and is given by

E = ks(t)k2 . (14)

• The correlation between two signals sk (t) and sm (t), which is a measure of the similarity between
these two signals, is given by

< sk (t), sm (t) > < sk (t), sm (t) >

ρkm = = √ . (15)
ksk (t)kksm (t)k Ek Em

• The distance between two signals sk (t) and sm (t), which is also a measure of the similarity between
these two signals, is given by
 p 1
2
dkm = ksk (t) − sm (t)k = Ek + Em − 2 Ek Em Re[ρk,m ] . (16)

If Ek = Em = E, then
1
dk,m = [2E(1 − Re[ρk,m ])] 2 . (17)

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