First-Order LTI System

Chapter 2: Principles of High-Speed Communications 10

2.3 Link Reliability

2.3.5 ISI Impact on BER

2.3.5.1 First-Order LTI System

If the link has a first-order system response with an associated time constant, τ^{, the} received pulse shape can be written as

P_e P a_k⋅p_o(T_s+(m–k)T_b) +n T( _s+mT_b)

k=–∞,k≠m

∞

∑

>¹2---

⎝ ⎠

⎜ ⎟

⎜ ⎟

⎛ ⎞

=

(2.13)

where we define that relates the system time constant and the bit period. The ISI term for a₀ can be calculated by replacing (2.13) in (2.8) for m=0 as

(2.14)

where the sum goes only to k=-1 because we assume the system is causal. T_s is the sam- pling time offset from t=0 and . The sampled value of the current symbol, i.e., a₀, can be calculated from the first term on the right in (2.8) as

. (2.15)

The optimum sampling point for the first-order system is at T_s=T_b because (2.15) reaches its maximum at this sampling point. Equation (2.14) demonstrates that the interference impact of the prior bits decreases exponentially. When the impact of only one prior bit, i.e., a_-1, is significant, ISI terms will be concentrated around two mean values, ISI₀ and ISI₁. The two mean values can be calculated from the expected value of ISI in (2.14) when it is conditioned on the value of a_-1. We have

(2.16)

(2.17) p_o ( )t

0 t≤0

1 e

t τ-- –

– 0≤ ≤t T_b 1

α---–1

⎝ ⎠

⎛ ⎞ e

t τ-- –

⋅ T_b≤t

⎩⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎧

=

α≡e^–Tb⁄τ

ISI a_k⋅p_o(T_s– Tk _b)

k= –∞ 1 –

∑

^α

T_s T_b ---

a_k⋅(1–α) α⋅ ^–k–1

k=–∞ 1 –

∑

= =

0≤T_s≤T_b

p T( )_s 1 α

T_s T_b ---

–

=

ISI₀ E ISI a{ _–1 = 0} 1 2---α

T_s T_b ---+1

= =

ISI₁ E ISI a{ _–1 = 1} α

T_s T_b ---

1 α

---2

⎝ – ⎠

⎛ ⎞

= =

where E{.} is the expected value. The two mean ISI terms perturb the amplitude at the sampling point. Because of the stochastic nature of the data, the ISI at any point can be modeled by a random variable. Then, the ISI distribution can be represented by two prob- ability mass functions, i.e., two delta functions at the values ISI₀ and ISI₁, with probability weight p and (1-p), respectively, where p is the probability of a_-1=0. The overall ampli- tude distribution can be found by the convolution of the ISI distribution and the Gaussian noise distribution as shown in Figure 2.14.

The optimum slicing threshold, V_TH, can be calculated from the average of the four possible mean signal levels in Figure 2.14, which simplifies to V_TH=0.5 independent of T_s. If the receiver input noise is white with double-sided power spectral density , the amplitude noise variance at the sampling point is reshaped by the first-order system transfer function. The total noise power can be calculated from

. (2.18)

Similar to Section 2.3.2, we can now calculate the total BER as

Figure 2.14: Total amplitude distribution at the sampling point when ISI impact of one bit is taken into account

ISI distribution

Noise

distribution Total distribution

* =

IS₀I ISI

1

po(T

s)+

IS₀I po(T

s)+

ISI₁

V_TH=0.5

N₀ ---2

σ²

N₀ ---2 1+τ²ω² --- fd

∞ –

∞

∫

^N---4τ⁰

= =

(2.19)

which can be evaluated for different sampling time by using (2.15)–(2.18). Figure 2.15 compares the BER at various signal-to-noise ratios (SNR) in the zero-ISI case in equation (2.6) with the BER in the ISI channels from equation (2.19), when the systems have the same noise bandwidth, i.e., equal σ. The BER curves are plotted for various 3dB band- width-to-bit rate ratios (BW/BR) for the ISI channel. The figure shows that the ISI degrades the performance of the link at large SNR values when the ISI dominates over noise. Also, as the bandwidth-to-bit rate ratio decreases, the BER degrades more.

BER 1

4--- Q 0.5–ISI₀ ---σ

⎝ ⎠

⎛ ⎞ Q 0.5–ISI₁ ---σ

⎝ ⎠

⎛ ⎞

⎝ +

= ⎛

Q p_o( )T_s +ISI₀ –0.5 ---σ

⎝ ⎠

⎛ ⎞ Q p_o( )T_s +ISI₁–0.5 ---σ

⎝ ⎠

⎛ ⎞

+ ⎠⎞

+

10 15 20

-15 -10 -5

0

BW/BR=0.5

BW/BR=0.75 Zero ISI

21 22 23 24

-14 -12 -10 -8

Figure 2.15: The BER vs. SNR for various normalized bandwidths compared to the zero-ISI BER of equation (2.19), sampled at optimum point, i.e., T_s=T_b

SNR [dB]

log

10

[BER]

Figure 2.16 relates the BER to the system 3dB bandwidth, f_-3dB, for various noise power spectral density, when the signal is sampled at the optimum point, i.e., T_s=T_b. Evidently, at very small f_-3dB, the ISI is severe and limits the BER. However as bandwidth gets excessively large, the noise power that is injected into the receiver is the dominant contributor to the link-quality degradation and causes higher BER. Consequently, there is a trade-off between the system noise and the ISI impact. There exists an optimum bandwidth that minimizes BER. The optimum bandwidth in the case of the first-order system is around 40% of the bit rate when T_s=T_b. In typical wireline link architectures the sampling clock is in the middle of the eye at T_b/2. Although this results in simple hardware implementations, T_b/2 is not necessarily the optimum sampling point. The BER for when sampling occurs in the middle of the eye, i.e., T_s=T_b/2, is plotted in Figure 2.17.

The same trade-off exists between the noise and the ISI. However, the optimum

0.2 0.4 0.6 0.8 1 1.2

-14 -12 -10 -8 -6 -4 -2

Figure 2.16: ISI and noise trade-off as normalized bandwidth variations justifies existence of a minimum for BER

log10[BER]

f_-3dB/Bit Rate

Ν₀^[V2/Hz]

5e-3 6e-3 7e-3 8e-3

bandwidth is now at around 70% of the bit rate which agrees with the well-known optimum bandwidth-to-bit rate ratio for best sensitivity in broadband receivers [36].

We also notice by comparing Figure 2.16 and Figure 2.17 that for equal input noise, the location of the sampling point affects the minimum achievable BER. In fact, the plot in Figure 2.16 can be reproduced for all possible sampling points, and the optimum sampling point can be determined from the plot that results in the smallest minimum-achievable BER. The optimum receiver bandwidth is also determined from the same plot. We will elaborate on this topic for the link design when we add the effect of jitter to the BER. We will analytically derive the two-dimensional BER contours that allow the designer to simultaneously determine the optimum bandwidth and the optimum sampling point to minimize the BER.

0.3 0.5 0.7 0.9 1.1 1.3 1.5

-12 -10 -8 -6 -4 -2

Figure 2.17: The optimum bandwidth for minimum BER when sampling point is in the middle of the eye at T_b/2

log10[BER]

f_-3dB/Bit Rate Ν₀^[V2/Hz]

3e-3 4e-3 5e-3 6e-3