Teaching Schedule

» Consider M-ary Digital Transmission

– Introduction
– Signal Space Concepts
– Basis Vectors/functions
– Determination of an orthogonal basis set
(Gram-Schmidt Orthogonalization)

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 Motivation
• Did we really obtain an optimal binary demodulator?
– We obtained an optimal threshold VT and optimal filter (Matched Filter).
– Does that really mean “optimal”?
– What we have done is only optimal w.r.t. the considered structure of the
receiver.
– However, why the optimal binary demodulator has to have such
structure?
• What about M-ary modulation?
– We could in fact increase the bit rate by transmitting more information
bits per modulation symbol.
– How to design the modulator and demodulator?
• How to compare different modulation schemes?
– Messy equations, difficult to obtain useful insights.

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 Signal Space Concepts and Signal
Representation
It turns out that the key to analyzing and
understanding the performance of digital
transmission is the realization that signals used in
communications can be expressed and visualized
graphically.

Thus
We need to understand signal space concepts as applied to
digital communications

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 Overall Objectives/Goals

• To analyze the problem of digital signal detection from a

fundamental point of view.

• To understand the digital modulation and demodulation from

a geometric perspective
– Easy to understand
– Useful design insights can be obtained without too much math
– Concept of Signal Space

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 Signal Space Concepts
• Signal space concepts will allow a more general way
of looking at modulation schemes.
• By choosing an appropriate set of axis for our
signal constellation, one can:
– Design modulation types which have desirable properties
– Construct optimal receivers for a given modulation technique
– Analyze the performance of modulation schemes using very general
techniques.

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 Concept of Signal Space

Time Domain
Representation s(t)

Geometric Domain
Representation s

Geometric Representation of Signals

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 Representation of Signals
• (1) Time Domain:
Signal is represented by a function in time, s(t).
Waveform (the shape of the function) could be observed.
Periodic Signal
Starting
Phase = 30
S(t + T) = s(t) for all t. T = period
Frequency (cycle per second) = 1/T (Hz).

Amplitude = A

Period = T
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 Representation of Signals
• Frequency Domain:
Signal could be represented by a function of frequency
S(f) as well.
For some aperiodic signals,
could be decomposed into components of “sin” and “cos”.
Each component has different (amplitude, frequency,
phase).

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Frequency Decomposition

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 Representation of Signals
• Geometric Domain (Signal space)
– Signal s(t) is represented as a “vector” s (with coordinates)
– For a vector to be meaningful, we need to define the space first
» What is the “frame-of-reference”?
» The “frame-of-reference” is defined by “x-axis”, “y-axis”,.....

• Geometric Domain (Signal Space) Define the “frame-of-reference”

– Signal could be represented by a point in a
space.
– Step 1: Given a set of M signals,
define a D-dim signal space with basis
.
– Step 2: Find out the coordinates of each
signals by:

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 Geometric Representation of
Signals
• Time Domain (x(t)), Frequency Domain (X(f)),
Geometric Domain (x) are just different views
looking at the same coin.
– The physical characterization of the coin will be the same no
matter you are computing from which domains
–
T
E = 0 |x(t)|2 dt (Time Domain Energy)
⇥
E = ⇥ |H(f )|2 df (Frequency Domain Energy)
E =< ⌦x, ⌦x >= ⇥⌦x⇥2 (Geometric Domain Energy)

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 Example 1
• Consider 4 signals

Find the orthonormal basis functions (orthonormal axis) of the

Signal Space that contains the 4 signals.

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Example 1

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 Example 1
x1 (t) = T /3 1 (t) +0 2 (t) +0 3 (t)

{x1 (t), x2 (t), x3 (t), x4 (t)}

Time Domain
Basis Function
{ 1 (t), 2 (t), 3 (t)}

Geometric Domain {x⇤1 , x⇤2 , x⇤3 , x⇤4 }

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 Example 2
Consider the following signal set:

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 Basis Functions

• By inspection, the signals can be expressed in terms

of the following functions:

• These are known as basis functions.

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Constellation Diagram

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 Signal Space and Basis
Functions

• Two entirely different signal sets can have the same

geometric representation.
• The underlying geometry will determine the performance
and the receiver structure for a signal set.
• In the previous examples, we were able to guess the correct
basis functions.
• In general, is there any method which allows us to find a
complete orthonormal basis for an arbitrary signal set?
– Gram-Schmidt Orthogonalization (GSO) Procedure

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 Vector Space
• A vector space V over a field F is a set of “abstract objects”
called “vectors”.
– The elements of V are called “Vectors”.
– The elements of F are called “Scalars”.
– Two basic “binary operations” (1) Vector additions; (2) Scalar Multiplications
that satisfy the following AXIOMS
» Associativity of Addition: u + (v + w) = (u + v) + w
» Commutativity of Addition: u + v = v + u
» Identity Elements of Addition: There exists 0 \in V s.t. 0 + u = u for all u \in V.
» Inverse Elements of Addition: For every v\in V, there exists -v \in V s.t. v + (-
v) = 0
» Distributivity of Scalar Multiplication (w.r.t. Vector Addition): a(u+v) = au +
av
» Distributivity of Scalar Multiplication (w.r.t. Field Addition): (a + b)u = au +
bu.
» Compatibility of scalar multiplication: a(bv) = (ab)v
» Identity element of scalar multiplication: there exists 1 \in F s.t. 1v = v for all
v \in V.

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 Vector Space Examples

• Coordinate Space over Real elements:-

– V = {(a1, a2, ..., an): ai \in R} a vector space can be composed of n-tuples of real
numbers. (Field = R)
• Coordinate Space over Complex elements:-
– V = {(a1, a2, ..., an): ai \in C} a vector space can be composed of n-tuples of
complex numbers. (Field = C)
• Function Space (Signal Space):-
– V = Functions from any fixed domain to F also forms a vector space.
– e.g. Functions of time --> R (signal space) is a vector space.

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 Inner Product Space
• A vector space (V,F) does not have notion of geometry (or
topology)
– Notion of distance? (Two vectors are close or far away from each other)
– Notion of topology? (open set, closed set, limits)
– Notion of geometry? (Circle??)
– All these requires “norm”
– Notion of angle? (angle between two vectors)
– All these requires “inner product”
• A vector space (V,F) with an “inner product” is called “inner
product space”
– Inner Product is a mapping <u,v>: V x V --> F that satisfy the following axioms
» <u,v> = <v,u>*
» <u+v, w> = <u,w> + <v,w>
» <au,v>=a<u,v>
» <u,u> >= 0 and <u,u> = 0 iff u = 0.

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 Geometric Concepts in Inner
Product Space
• Length of a vector:
– ||v||2 = <v,v>
• Distance between two vectors:
– ||v-w||2=<(v-w),(v-w)>
• Angle between two vectors:
<v,w>
cos ✓ = kvkkwk
• Orthogonal vectors: <v,w> = 0
• Circle (xc, r): ||x - xc || = r
• Limit of a sequence:
limn!0 vn = v
For any ✏ > 0, there exists n0 such that for all n > n0 , kvn vk < ✏

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 Vectors and Space Concepts
• An n-dimensional space S is defined by a set of n basis
vectors (e1, e2, … en);
– S = span (e1, e2, … en);
⇒Any vector a can be written as

n = dimension = maximum number of linearly independent vectors in the

vector space

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• Notation: Coordinate Representation of vector a.

• Definitions:

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 4) A set of vectors are orthonormal if they are mutually
⊥ and all have unity norm.

• 5) A transformation h(⋅) is said to be Linear if

∀α, β ∈ IR and ∀a and b.

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6) a1, a2, …, an are independent if none of these vectors
can be written as a linear combination of the others.

7) Triangular Inequality:

For any vectors a and b,

With equality iff

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8) Cauchy – Schwartz Inequality:

9) Pythagorean Relation
if a and b are ⊥

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 Basis Vectors

• Let (a1, a2, …, an) be a set of n vectors. These vectors

are independent if it is impossible to find constants α1,
α2, …αn (not all zero) such that

• In an n-dim space, we can have at most n independent

vectors

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 Signal Space Concepts
• Basic Idea: Any entity that can be represented by n-tuple
is an n-dim Vector ⇒ If a finite-duration signal (Ts) can be
represented by n-tuple, then it is a vector.

• Let ϕ1(t),ϕ2(t),….,ϕn(t) be n finite duration signals (Ts)

• Consider a finite-duration signal x(t) and suppose that

• If every signal can be written as above ⇒ {ϕ1(t),ϕ2(t),

….,ϕn(t)} ~ basis and have n-dim space

x(t) , x = (x1 , ..., xn ) with respect to basis {'1 (t), ....'n (t)}
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 Ts
• Define “dot-product” as < x(t), y(t) >= x(t)y (t)dt
0

• Basis set {ϕk(t)}n is an orthogonal set if

• If kj≡1 ∀j ⇒ {ϕk(t)} is an orthonormal set. In this case,

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 Key Property
Given a signal space S = span{'1 (t), ....'n (t)} and a finite duration signal
x(t) 2 S
(1) Computing Dot-Product
Let x(t), y(t) 2 S, x(t) , x = P(x 1 , .., xn ), y(t) , y = (y1 , .., yn ). For
n
orthonormal basis, < x(t), y(t) >= i=1 xi yi

(2) Energy of x(t)

R Ts
Es = 0
|x(t)|2 dt (Time Domain Method)

R1
Es = 1
|X(f )| 2
df (Frequency Domain Method)

Es = kxk2 =< x(t), x(t) > (Geometric Domain Method)

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Geometric Domain Representation
• Geometric Domain (Signal Space)
– Signal could be represented by a point in a
space.
– Step 1: Given a set of M signals,
define a D-dim signal space with basis
.
– Step 2: Find out the coordinates of each
signals by:

• Question 1) How to find the signal space

(basis signals) that contains {s1 (t), ..., sM (t)}
• Question 2) How to find the coordinate of
each signal?
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 Step 1) Gram-Schmidt
Orthogonalization for Vectors
• Given a set of M vectors , the G-S procedure allows one to find out the
“orthonormal basis” of the signal space (with the minimum dimension) to
contain all the M vectors.
Projection of x⇥m on the current vector
– Step 1: space spanned by { ⌅1 , ..., m⌅ 1 }

– Step 2:

– Step m:

The process continues until m=M or

• Similarly, for signal space, vector = signal.

– Given a set of M “signals” (vectors), we can use the same GS procedure to find out the “orthogonal
basis” (basis signals) of the signal space (with min dimension) to contain all the M signals.
– Use the same procedure except with the understanding that

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 Summary of GSO

• 1st basis function is a normalized version of 1st signal.

• Remaining basis functions are found by removing portions
of signals which are correlated to previous basis functions,
and normalizing the result.
• This procedure is repeated until all basis functions are
found.

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 Step 2) Computing the
Coordinates
Given the orthonormal basis { 1 (t), ..., D (t)} that contains the M finite
duration signals {s1 (t), .., sM (t)},

si (t) , si = (si1 , ..., siD )

R Ts ⇤
sij =< si (t), j (t) >= 0
si (t) j (t)dt

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Example

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 Example

c. Draw the constellation diagram for the signals

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