The Butterworth filter is designed to have a frequency response that is as flat as mathematically possible in the passband. Butterworth has a slower roll-off, and thus require a higher order to implement a particular stopband specification. However, Butterworth filter has a more linear phase response in the passband than the Chebyshev and elliptic filters. Eq. (4-10) expresses a general form of a filter design.
⎭⎬
butterwort (4-10)
A 5th order of Butterworth filter is designed in this example so that the variable n in Eq. (4-10) is replaced with the number five. As a result, the corresponding transfer function has the following polynomials.
5
This s-domain representation gives an impulse response as depicted in Fig. 4-6. A Butterworth does not guarantee a zero-crossing at a T-spaced location other than the main signal tap. So, an additional ISI is introduced in this filter, although the passband response performs linear and minimum ripple behaviors.
0 2 4 6 8
Impulse response of a butterworth function
Time index (nT)
Amplitude
butterworth function
Fig. 4-6. Butterworth function impulse response
In the following, the filter responses take the mentioned sinc, raised-cosin, and 5th-order Butterworth filters for illustration. The DSTC algorithm is investigated from the main-tap and side-tap of those filters. The signal main-tap to the side-tap ratio is defined and maximized a
) signal-to-ISI power ratio (SIR). Thus, the εopt is determined when the minimum ISI power sum appears in the denominator of Eq. (4-12). In other words, the SIR of the sampled signals becomes maximized when the optimum sampling instance is determined. Here, f(t) is replaced by the filter impulse responses as shown in Figs.
4-4~4-6. A non-calibrated sampling timing error may yield low signal-integrity data even in the absence of noise, implying there is system performance degradation when sampling time is not well-calculated. The SIR response depends on the filter responses. Figs. 4-7~4-9 demonstrates the corresponding SIR curves from those corner filters (sinc, raised-cosine, and Butterworth filters). It is found that a sharper frequency response implies a better SIR curve performance. This is due to the faster convergence of the impulse response signal tails appeared as the side-lobes. In other words, fewer impulse tails contributes fewer interference taps. Consequently, a correct sampling position would provide excellent signal quality and overall performance. However, a sharper frequency response of a filter (implying fewer impulse response tails) requires higher order of a filter design, and thus requires high design cost. Accordingly, this SIR may be viewed as a design indicator that the system tradeoff between overall performance and filter design could be discussed in this way.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 100
101 102 103
104 Signal-to-ISI Ratio v.s. Sampling Timing Error
Sampling Timing Error: ε
SIR (dB)
sinc function
Fig. 4-7. The signal-to-interference-ratio (SIR) for a sinc function impulse resopnse
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 100
101 102 103
104 Signal-to-ISI Ratio v.s. Sampling Timing Error
Sampling Timing Error: ε
SIR (dB)
Fig. 4-8. The signal-to-interference-ratio (SIR) for a raised-cosine filter impulse response
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 100
101 102 103
104 Signal-to-ISI Ratio v.s. Sampling Timing Error
Sampling Timing Error: ε
SIR (dB)
Butterworth filter
Fig. 4-9. The signal-to-interference-ratio (SIR) for a 5th-order Butterworth filter impulse response
To find the optimum sampling time εopt, the ratio given by Eq. (4-12) cannot be calculated directly because both the fT(t) and the f(t)= fT(t)∗fR(t) are unknown to any receivers. Therefore, an alternative approach, the maximum absolute-squared sum equivalent to Eq. (4-12) and also hardware realizable, is examined. Accordingly, the maximum absolute-squared-sum (MASS) of the received signals is
2
The expression is further described as
⎭⎬
where wB;ε is the band-limited zero-mean additive noise sampled at timing offset ε
with wB= w(t)∗fR(t). The expectation of Eq. (4-14) is
{ } { }
Notably, wB;ε is assumed to be independent of transmitted signals, and the noise is a zero-mean random process with E
{
wB;ε[n]}
=0. So, the expectation of the cross term becomes zero.{ }
operation suppresses the CFO factor. Therefore, the expected received signal power is composed of the transmitted signals filtered by the f(t) and the band-limited noise power. The effects of f(t) on the transmitted signals are expressed as main signal taps| 0
E Tε may be assumed to be a constant, say unit power, because every received signal power is adjusted by applying an automatic gain control (AGC) mechanism, thus normalizing the signal power to the dynamic range of the ADC. For simplicity, E{|xT;ε[n]|2|}=1 is defined. Accordingly, Eq. (4-17) becomes
{ }
Thus, Eq. (4-18) gives the summation of signal power, interference power, and cross-correlation term. This cross-correlation term can be expressed as
{ }
Since XT is the output symbol of BPSK, QPSK, or QAM, which is assumed to be i.i.d.
variables with zero mean. The expectation { [ ] [ ] }
These information symbols are assumed to be independent, and then
=
Consequently, the expected absolute-squared value of the received signals is determined by the power of both the filter response and AWGN. Based on the SIR definition, Eq. (4-20) is rewritten as
2
where
∑
≠
=
0
|2
] [
|
m
m f
Iε ε is the interference power of the filter tail. A characteristic function (CF) of E{|xR;ε[n]|2} is defined as
CF=(SIRε + )1 ⋅Iε (4-22)
A sharper CF curve is more easily recognized to calibrate the sampling timing errors.
Fig. 4-5 plots a CF curve that corresponds to the raised-cosine filter of Fig. 4-4 in a noiseless channel. This finding reveals that the maximum E{|xR;ε[n]|2} implies the optimum sampling instance εopt. Therefore, the εopt search based on the SIR curve in Eq. (4-12) is transferred to the search of the maximum E{|xR;ε[n]|2}, i.e.
})
| ] [ {|
arg(maxE xR; n 2
opt ε
ε = (4-23)
According to the provided reference filter responses, the calculated absolute-squared sum of received signals in variant sampling offset can be depicted as following. The CF curves are the indicators for a receiver that the best sampling instance in each sample period is easy or difficult to identify. This indicator should be considered together with the SIR curve. A sharper SIR curve does not necessarily corresponds to a sharper CF curve. The design constraints are provided that the SIR is desired to be as flat and high as possible whereas the CF is sharper as possible. A sharper CF curves enables accurate calculation of best sampling instance for a receiver, and a high and flat SIR curve implies a miss-calculation of best sampling instance does not degrade the overall system performance too much. Figs. 4-10~4-12 illustrate the CF curves from different impulse responses. The proposed MASS calculation for the best sampling instance requires a sharper CF curve that is designed by the overall consideration of the system building block circuits. Moreover, the sharper CF curve
has larger noise immune capability that prevents the MASS computation from the disturbance from the noise.
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 0.75
0.8 0.85 0.9 0.95 1 1.05
Sampling Timing Error: ε
Characteristic Function
sinc function
Fig. 4-10. Characteristic function of a sinc function
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 0.5
0.6 0.7 0.8 0.9 1
Sampling Timing Error: ε
Characteristic Function
roll-off factor(β) = 0.2 roll-off factor(β) = 0.5 roll-off factor(β) = 0.8
Fig. 4-11. Characteristic function of a raised cosine filter
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 0.5
0.6 0.7 0.8 0.9 1
Sampling Timing Error: ε
Characteristic Function
Butterworth
Fig. 4-12. Characteristic function of a 5th-order Butterworth filter