EQUALIZATION - Advanced Signal Integrity for High-Speed Digital Design - page 496

Digital data Digital data

Analog signal

Noise
n ( t )

T

Channel
h c ( t )

Transmitter
h t ( t )

Receiver
h r ( t )

Detector

Σ

x ( t ) { x ' k }

{ x k } y ( t )

Figure 12-1 Communications channel view of a high-speed signaling interface.

1/ f

Periodic
Signal

T symbol T symbol

Random
Data
Signal

1 0 0 0 1 1 0 1 0 1 1

Figure 12-2 Symbol rate illustration for a binary NRZ signal.

bandwidth of the signal is equal to the repetition frequency. † Thus, we have two
symbols per cycle.
One of the outcomes of Shannon's work states that the maximum number of
bits per symbol, B , that can be transmitted without error is given by

2 log 2 1

1 P s

P n

B = + (12-3)

where P s is the average signal power and P n is the noise power. The quantity
P s /P n is also known as the signal-to-noise ratio (SNR). Equation (12-3) assumes
that the noise is Gaussian, meaning that it is constant at all frequencies within
the channel bandwidth, which is a reasonable approximation for digital systems
[Sklar, 2001].
Combining the preceding equations gives the Shannon-Hartley theorem ,
which expresses the maximum data transfer rate in bits per second (b/s) as a
function of the interconnect channel bandwidth and the SNR:

D = BWlog 2 ( 1 + SNR ) (12-4)

Equation (12-4) shows that we can increase throughput across an interchip
interconnect either by increasing the signal-to-noise ratio or by increasing the

† A real digital signal contains energy at harmonic frequencies above the fundamental, which we can
estimate from the rise time (BW = 0 . 35 /t r ) as derived in Section 8.1.3. In this analysis, however,

we are considering only the fundamental frequency.