Quadrature Amplitude Modulation

Forrest Sedgwick

UC Berkeley EECS Dept.

EE290F

October 2003

Analog vs Digital

Information Theory vs Signal Analysis Discrete Levels vs Analogous Representation

Sacrifice arbitrarily precise representation of signal Gain arbitrary degree of reproducibility of given signal

KEY BENEFIT Discrete information can be transmitted with arbitrarily low

error rates EVEN ON A NOISY CHANNEL Digital information content measured in units of bits,

decimals, or nats

Shannon’s Channel Capacity

Channel capacity C (bits/sec) is the speed at which information can travel over a channel with an arbitrarily low error rate i.e. when a system is transmitting bits at or below C then for any BER e>0 there exists a code with block length n which will provide a BER < e.

WN

PWC

0

1logAssumes noise is thermal – Gaussian and White

www-gap.dcs.st-and.ac.uk/~history/ Mathematicians/Shannon.html

Modulation

All channels consist of some continuous parameter Must map discrete states onto continuous property Must have a decision circuit to map the state of the

modulated channel into a discrete state As number of levels or states M the behavior of

the digital system does not approach that of an analog system, due to the decision circuit

Number of Levels

Digital communications relies on a finite number of discrete levels

Minimum number of levels is two (binary code) Shannon Capacity helps determine optimum number

of levels for a given bandwidth, SNR, and BER

0

1logN

Err b

W

Cr

Limits on Communication Channels

Two types of communication channels

r<<1 – Power Limited High dimensionality

signaling schemes Binary

r>>1 – Bandwidth Limited Low dimensionality Multilevel

0

1logN

Err b

Proakis and Salehi, pp. 738

Modulation Scheme

A channel with lowpass frequency characteristics is called baseband. Digital information is transmitted directly

Ex. Pulse Amplitude Modulation (PAM) A channel far removed from DC (like optical) is called

a bandpass channel Transmission on a bandpass channel requires

modulation of a carrier Amplitude Shift Keying (ASK) Phase Shift Keying (PSK) Frequency Shift Keying (FSK Quadrature Amplitude Modulation (QAM)

Amplitude Shift Keying (ASK)

Amplitude of carrier wave is modulated Equivalent BER vs SNR to baseband PAM

Proakis and Salehi, pp. 306

Angle Modulation (PSK and FSK) Frequency is time derivative of phase, PSK

and FSK are somewhat equivalent

Proakis and Salehi, pp. 332

PSK: Digital Angle Modulation Usually in digital communications PSK is chosen over

FSK Easier to create multilevel codes Possibility of using differential phase shift keying (DPSK)

Uses phase shifts relative to previous bit Eliminates need for local oscillator at receiver Use Gray Code to minimize effect of errors

Proakis and Salehi, pp. 631

Quadrature Amplitude Modulation

Amplitude and Phase of carrier are modulated

Discrete amplitudes and phases form a constellation

Can also think of QAM as a “complex” amplitude modulation scheme

Proakis and Salehi, pp. 653

Constellations

Different constellations require different SNR for a given BER

(d) is lowest power by about 1dB (for given BER)

(a) and (b) are rectangular Rectangular constellations

offer very simple modulation/ demodulation schemes

ASK two quadrature carriers - same frequency but 90 out of phase

Mix quadrature carriers for output

Proakis and Salehi, pp. 653

QAM vs ASK (multilevel)

QAM has a tremendous advantage in noise performance Energy in every bit (including zero) Substantially more complex (coherent detection vs photodiode)

Proakis and Salehi, pp. 565 Proakis and Salehi, pp. 495

QAM vs PSK

4-QAM and 4-PSK have same power penalty

For k>4, k-QAM is an improvement over k-PSK

Proakis and Salehi, pp. 639

Applications of QAM

Used in bandwidth-limited applications Modems: telephones have 3kHz bandwidth, excellent SNR

(20dB) => M-ary QAM Cellular Telephones: Bandwidth is at a premium, very

expensive (However, POWER is also at a premium...)

Limitations

Almost always requires a highly stable local oscillator In the optical domain this is very expensive Possible (but difficult) to use differential phase keying Performance limits still not reached for

Direct detection Signal Dimensionality (DWDM) Transmitter Power

References

John G. Proakis, Masoud Salehi, Communications Systems Engineering, Prentice Hall 1994