Amplitude Modulation: Digital QAM The goal of quadrature sampling is to be able to digitize an analog bandpass

signal It is a form of digital modulation where the digital information is obtained in

both the amplitude and phase of the transmitted carrier.

Quadrature Sampling

Bit rate rb=rlog2MTransmission Bandwidth

BT=r=rb

log2 MEfficiency rb

BT

=log2 M

Symbol Duration

D=1r=log2M

r b

Binary Quadrature Amplitude Modulation

8-QAM Transmitter

8-QAM Reciever

16-QAM

Frequency Shift KeyingA type of modulation which converts digital signals via discrete frequency changes in the carrier signal

2-ary FSK (BSFK) Two carrier frequencies, f1 and f2, represent binary 1 and 0.

BSFK: Fourier Transform

BSFK Bandwidth (Carson’s Rule) : B=2 (∆ f + f )

Continuous Phase FSK (CPFSK) Due to the 1/Tb restriction, the signal has a continuous phase in inter-bit

transitions. f should be a multiple of 1/Tb, so that starts and ends at the same pointBFSK Generation:

Switching modulation - switch Product modulation - diode + multiplier Voltage-controlled oscillator

BFSK Detection: Coherent detection

o Carrier phase must be continuouso multiplier + LPD + Added + Level Converter

Non-coherent detectiono No synchronizationo BFS + Envelope detectors + sample and hold + Level Converter

M-ary FSK Equations:General equation s ( t )=A cos (2π f it)equation for the frequencies (fi) f i=f c+

2 iT

Fc = center frequencyI = ith frequency

BandwidthBT=

2MT

T = symbol duration

MFSK Generation: series to parallel conv + 2x4 Decoder + multiplier + adderMFSK Detector: multiplier + LPF + decoder + Parallel to Series converter

Phase Shift Keying Involves the carrier signal changing between phases determined by the logic

states of the input bit stream Change the phase of the sinusoidal carrier to indicate information

Resulting Phase Shifts

Bits per symbol of M-aryPSK

COHERENT - phase is imposed and measured with respect to a fixed carrier of known phase.

DIFFERENTIAL - phase change between two consecutive symbols

BPSK Generation

BPSK Detection

QPSK Equation

QPSK Generation

QPSK Detection

8PSK Modulation

FINITE FIELDS AND GALOIS FIELDSProperties of Finite Field

- Convention : f(x) = fyxy + f(y-1)x(y-1) + … + f1x + f0

- Finite Field - a field with a finite amount of elements.

- Elements : - The number of elements in a Finite Field must be a prime power, say q =

pm, where p is prime.Primitive Elements

- An element whose powers constitute all the non-zero elements of the fields (e.g. in GF(7), primitive element = 3)

- Let a= primitive element, multiplicative inverse aj is:

- ; Divisibility and Roots

- For an element in the field, a, if it is a root of the polynomial such that f(a) = 0 then the polynomial is divisible by x-a.

Irreducible Polynomials- f(x), a polynomial of degree m over GF(p) is said to be irreducible over

GF(p) if f(x) is not divisible by any polynomial over GF(p) of degree less than m but greater than zero.

Primitive Polynomials of GF(2 m ) - f(x), a polynomial of degree m, is said to be primitive if: 1. Irreducible

and monic (coefficient of highest degree is 1) 2. The smallest possible

integer n for which f(x) divides xn−1where n=2m+1 and m

is the degree of the polynomial- Table of Primitive Polynomials

m Primitive Polynomials

m Primitive Polynomials

2 X2+X+1

7

X7+X+1X7+X3+1X7+X3+X2+X+1X7+X4+1X7+X4+X3+X2+1X7+X5+X2+X+1X7+X5+X3+X+1X7+X5+X4+X3+1X7+X5+X4+X3+X2+X+1X7+X6+1X7+X6+X3+X+1X7+X6+X4+X+1X7+X6+X4+X2+1X7+X6+X5+X2+1X7+X6+X5+X3+X2+X+1X7+X6+X5+X4+1X7+X6+X5+X4+X2+X+1X7+X6+X5+X4+X3+X2+1

3 X3+X+1X3+X2+1

4 X4+X+1X4+X3+1

5

X5+X2+1X5+X3+1X5+X3+X2+X+1X5+X4+X2+X+1X5+X4+X3+X+1X5+X4+X3+X2+1

6

X6+X+1X6+X4+X3+X+1X6+X5+1X6+X5+X2+X+1X6+X5+X3+X2+1X6+X5+X4+X+1

Galois Field GF(2 m ) - It begins with the two elements from GF(2), 0 and 1 and introduce a

new symbol α which is a primitive element of the field GF(2m).

- Note: GF(2m) { 0, 1, 2, 3, …, 2m-1}; GF(2m) = { 0, 1, α, α2, α3, ... αj }

- GF(2m) = { 0, 1, α, α2, α3, ... α 2m−2 }; The set GF(2m) is a Galois field of

2m elements.

Making GF(24 )- Since m=4 form the table of primitive polynomials

- Set f ( α )=α 4+α+1=0 then α 4=α+1α=α α 2=α 2α 3=α 3α 4=α+1α 5=α 2+α+1

…Roots of Polynomials

- Polynomials may have roots which are not from their field but from their extension fields (not from GF(2) but from GF(2m)

- If β is an element and root of an extension field, β2l

is a root of f(x)

and also a conjugate of βMinimal Polynomials

- The minimal polynomial ϕ (x ) of an element β of a Galois Field is

the product of x−β and every x−β2l

.

- If β is a root of f (x) then f ( x )is divisible by ϕ ( x ) .LINEAR BLOCK CODESTerminology ( k=message length, n=codeword ,

q=n−k=chkbits¿

Block code (n , k ), code efficiency RC=kn

Distance – (# of different bits on same position), weight =no. of non zero bitsCyclic Codes – all cyclic shift of a codeword is also a codeword, all zero codeword is included in C, and sum of any two other codewords is also a valid codeword.NonSystematic

- Generator Matrix for nonsystematic (same with primitive polynomial)

- ascending order then convert to binary, r=n−k , n × n matrix

- ENCODING X=MG, X is the transmitted codeword;

xn−1=g ( x ) h ( x ) .to get the check matrix h, divide

xn−1by g ( x ).- Parity ¿̌

- Descending order then convert to binary, (n−k ) ×n matrix

- DECODING S=Y HT , construct syndrome table based on HT

Systematic – obtain Cyclic generator matrix by making (G=[ P∨I k ])- Add vector to obtain a vector with desired identity

- X=MG, code word is composed of parity bits then message

- Check Matrix H : h ( x )→ H [ PT|I ]→ HT

- Decoding : S=Y HT (syndrome table construction is the same)

Perfect Codes

- Codes- C (n , k ,d ) ,where d is the min. hamming distance.

- Code C can DETECT up to t errors: d ≥ t+1 ;- Code C can CORRECT up to t errors ; d ≥2t +1

- Hamming bound : M∑d=0

t

(nd )(q−1)d≤ qn where

M=2k

- For q=2: M∑d=0

t

(nd )=2n

Check Sum- Fixed sized datum computed from an arbitrary block of digital data.- E.g. Parity Bits, Cyclic Redundancy Check- Parity Checking – adds on bit to the pattern and requires modulo 2-

sum of all the bits of the pattern & the parity have defined answer

o Even (∑ ¿0¿ ,odd (∑ ¿1)Parity Checking using Square Arrays

- K message bits in square array. Rows & columns are checked by

2√k parity bits.Burst Error Detection using Parity bits

- Divides the message in frames then put the parity. Transmit many bits together just one from each frame

Cyclic Redundancy Check- it is characterized by specification of generator polynomial used as the

divisor in a polynomial long division over a finite field, taking the input data as the dividend (stuffed with (highest degree gen. polynomial) # of 0’s), and where the remainder becomes the result.

- Standard Polynomials for G(x)

HAMMING CODEHamming Code

- Uses multiple parity bits to be able to do single error correction, double error detection (SECDED)

- Each message or data bits gets checked by at least 2 parity bits

- (n , k ) where n = total # of bits, k = message length, q=n−k- Must satisfy 2q ≥ q+k+1- Systematic (message before parity or vice versa); Nonsystematic (no

particular order)(7,4) Binary Hamming Code

- Given 4 data bits (d1 , d2 , d3 , d4 ¿parity bits can be defined as:

-- ENCODING

o Represent each data bits and parity (depending on the data bit they’re checking) with a column vector

o →

generator matrix G (4

x7)

o X=MG- Decoding →based from the arrangement of the parity and data bits.

- ERROR? Construct H

- Get transpose of H and multiply by the received codeword. The S,

result, is the position of error (based from HT ¿- NON SYSTEMATIC

- ERROR CORRECTIONo Received : 1010111o Add the received parity and the calculated parity based

from the data received. The result is the position (p3p2p1)

- USING GENERATION MATRIX

- ENCODING

- DECODING