This is a handy presentation consisting the graphical and geometrical representation. Describing about orthonormality in a brief, along with basic vector and signal space. Also describing the QPSK constellation diagram and types of QPSK.

1. Signal Constellation,
Geometric Interpretation of
Signals
Group: 9
Ayan Das 11500320073
Arnab Paul 11500320075
Arnab Chatterjee 11500320076
Arijit Dhali 11500320078
Digital Communication & Stochastic Process
EC 503

2. Contents
1. Introduction
2. Geometrical Representation of signal
3. Constellation diagram
4. QPSK and its Constellation Diagram
5. Types of QPSK
6. Conclusion
7. References

3. Introduction
The essence of geometric representation of signals is to represent any
set of M energy signals{si(t)} as linear combinations of N orthogonal
basics functions, where N<=M. That part is to say, given a set of real-
valued energy signals s1(t),s2(t),….,sM(t), each of duration T second, we
write
Si(t) =σ𝑗=1
𝑁
𝑠ijɸj(t), ቊ
0 ≤ 𝑡 ≤ 𝑇
𝑖 = 1,2, … , 𝑀
Where the coefficients of the expansion are defined by:
Sij =‫׬‬
0
𝑇
𝑠i(t)ɸj(t) dt, ቊ
𝑖 = 1,2, … . , 𝑀
𝑗 = 1,2, … . , 𝑁

4. Geometrical
Representation of Signals
Derivation of geometrical representation of signals.
Combination of two orthonormal basis functions Φ1(t) and Φ2(t).
 Φ1(t) and Φ2(t) are orthonormal if: -
 ‫׬‬
0
𝑇𝑏
𝛷1(𝑡)𝛷2(𝑡) 𝑑𝑡 = 0
 ‫׬‬
0
𝑇𝑏
𝛷1
2
𝑡 = ‫׬‬
0
𝑇𝑏
𝛷2
2
𝑡 = 1 (Normalised to have unit energy).
 The representations are: -
 𝑠1 𝑡 = 𝑠11𝛷1 𝑡 + 𝑠12𝛷2 𝑡 ,
 𝑠2 𝑡 = 𝑠21𝛷1 𝑡 + 𝑠22𝛷2 𝑡 .
where 𝑠𝑖𝑗 = ‫׬‬
0
𝑇𝑏
𝑠𝑖 𝑡 𝛷𝑗 𝑡 𝑑𝑡, 𝑖, 𝑗 𝜖 {1,2}
 ‫׬‬
0
𝑇𝑏
𝑠𝑖 𝑡 𝛷𝑗 𝑡 𝑑𝑡, is the projection of 𝑠𝑖 𝑡 onto 𝛷𝑗 𝑡 .

5. Geometrical Representation of Signals
Basis Vectors:
The set of basis vectors {e1, e2, …,en} of a space are chosen such that: Should be
complete or span the vector space: any vector can be expressed as a linear
combination of these vectors. Each basis vector should be orthogonal to all
others
• Each basis vector should be normalized.
• A set of basis vectors satisfying these properties is also said to be a complete
orthonormal basis
• In an n-dim space, we can have at most n basis vectors
Signal Space:
If a signal can be represented by n-tuple, then it can be treated in much the
same way as a n-dim vector.
Let φ1(t), φ2(t),…., φn(t) be n signals
Consider a signal x(t) and suppose that If every signal can be written as
above ⇒ ~ ~ basis functions and we have a n-dim signal space
Orthonormal Basis:
In mathematics, particularly linear algebra, an orthonormal basis for an
inner product space V with finite dimension is a basis for V whose vectors
are orthonormal, that is, they are all unit vectors and orthogonal to each
other. For example, the standard basis for a Euclidean space Rn is an
orthonormal basis, where the relevant inner product is the dot product of
vectors. The image of the standard basis under a rotation or reflection (or
any orthogonal transformation) is also orthonormal, and every
orthonormal basis for Rn arises in this fashion.

6. Constellation
Diagram  It is the graphical representation of the complex
envelope of each possible state.
• The x-axis represents the in-phase
component and the y-axis the quadrature
component of the complex envelope.
• The distance between signals on a
constellation diagram relates to how
different the modulation waveforms are
and how easily a receiver can differentiate
between them.
it(t) qt(t)
-i0 +i0
+q0
-q0
I
I
it(t) = ± i0 cos(2𝜋fot)
qt(t) = ± q0 cos(2𝜋fot)

7. QPSK & Constellation
Diagram
Quadrature Phase Shift Keying (QPSK) can be interpreted
as two independent BPSK systems (one on the I-channel
and one on Q), and thus the same performance but twice
the bandwidth efficiency.
Carrier Phases:
{0 ,
π
2
, π ,
3π
2
}
Carrier Phases:
{
π
4
,
3π
4
,
5π
4
,
7π
4
}
Q Q
I
I
Quadrature Phase Shift Keying has twice the bandwidth
efficiency of BPSK since 2 bits are transmitted in a single
modulation symbol.

8. Types of QPSK
 Conventional QPSK has transitions through zero (i.e., 1800 phase
transition). Linear amplifiers are required highly.
 In Offset QPSK, the phase transitions are limited to 900, the
transitions on the I and Q channels are staggered.
 In /4 QPSK the set of constellation points are toggled each symbol,
so transitions through zero cannot occur. This scheme produces the
lowest envelope variations.
 All QPSK schemes require linear power amplifiers.
π
4
QPSK
Offset QPSK
Conventional QPSK
Q Q Q
I
I
I

9. Conclusion
 In this modulator, analog components like local oscillator and mixer
are completely eliminated which are frequency and temperature
sensitive .
 Here all the functions are performed by single FPGA. So the
limitations of modulator are completely removed from satellite
communication.

10. References
 https://www.testandmeasurementtips.com/what-is-orthogonal-
part-3-signal-constellations-faq/
 https://www.rcet.org.in/uploads/files/LectureNotes/ece/S6/Digital%2
0communication/UNIT-4.pdf
 https://en.wikipedia.org/wiki/Orthonormal_basis
 https://www.slideshare.net/gongadi/design-and-implementation-of-
qpsk-modulator-using-digital-subcarrier-18370723
 https://dsp.stackexchange.com/questions/74713/geometric-
representation-of-a-signal-using-basis-functions
 http://gn.dronacharya.info/EEEDept/Downloads/subjectinfo/Digital_C
ommunication/Unit-1/Lecture-6.pdf

11. THANK
YOU! Group 9