The proposed DSTC and PTCG [38] are evaluated in a multi-band OFDM (MB-OFDM)-based UWB system [2] with a low-density-parity-check (LDPC) code for error correction [54]. The signal bandwidth is 528MHz with QPSK and OFDM modulations, and the maximum data rate 480Mbps is selected in the following simulations.
The frame format in each packet can be divided as the packet synchronization sequence (Packet Sync. Seq.), the frame synchronization sequence (Frame Sync. Seq.), the channel estimation sequence (Channel Est. Seq.), and the data OFDM symbols.
Each OFDM symbol is transformed by 128-point inverse discrete Fourier transformation (IDFT). There is a time slot between every two OFDM symbols that does not transmit any information. This silent slot is used for each OFDM symbol to
change its transmission band frequency to another one. Accordingly, any change of controlling can be updated during this silent period as shown in Fig. 4-13.
Band Group N
Band Transition Duration Packet Sync Seq.
Frame Sync Seq.
3 OFDM Sym. Channel Est. Seq.
6 OFDM Sym. Data OFDM Sym.
Band Group N+1
Band Group N+2
Dynamic Sample-Timing Control
Phase-Tunable Clock Generator
Fwrd Bwrd
Fig. 4-13. Packet frame and waveform property for the DSTC operation
The dynamic timing-recovery starts the εopt search right after a packet is detected. Each packet is composed of 21 OFDM symbols at the beginning of each preamble frame (Packet Sync Seq), which is applied to the DSTC as shown in Fig.
4-14. With those 21 identical OFDM symbols in the packet sync sequence, each of which gives an absolute-squared sum, and the sampling time ε is changed in the time slots between OFDM symbols. In other words, the PTCG changes its output clock phase only during the time slots associated with band transitions such that signals in each OFDM symbol are sampled with the same clock phase within an OFDM block period as depicted in Fig. 4-14. This operation of DSTC corresponds to the flow shown in Fig. 4-15. In the band transition duration, the main function performs the PTCG update and sampling-instance shifting convergence. After the DSTC search process, the baseband starts the nominal signal synchronization and data decoding procedure.
Phase-Tunable Clock Generator
Band N
Band Transition Packet Sync Seq.
Frame Sync Seq.
3 OFDM Sym. Channel Est. Seq.
6 OFDM Sym. Data OFDM Sym.
Band N+1
Band N+2
Band N
Band N+1
Band Dynamic Sample- N+2
Timing Control
Fwrd Bwrd
]
; [n xRε
) (t xR
ε= ε1 ε2 ε3 ε4 ε5 ε6
Fig. 4-14. Packet frame used for the DSTC computation
Packet detection
Start DSTC search
Update PTCG
Parameter Update
Band slot finish
Timing Sync
Frequency Sync
Channel Equalization
Data Decoding
Band Transition
Dutation
Fig. 4-15. The operation flow for the DSTC computation
The sampling error phenomenon is revealed only if the waveform is depicted in an analog-like method so that the sampling instance can be slightly shifter forward or backward to simulate the sampling uncertainty behaviors. In this part, the simulation flow and methodology for this DSTC algorithm are illustrated, as shown in Fig. 4-16.
This also simulates and reflects the behaviors from the actual circuit operations.
First, the samples after the baseband circuits, before the digital-to-analog conversion (DAC), are generated that constitute the primitive symbol stream. In actual circuit operations, the output of the baseband modulator is generated by the flip-flop circuits, so the symbol value lasts until next latching clock comes. Accordingly, each symbol is up-sampled to simulate the latch-and-hold phenomenon, as depicted in Fig. 4-16(b).
Before the baseband digital signals are converted and transmitted to the receiver side, the equivalent filter response, including the DAC, RF-TX, channel, RF-RX, and the ADC circuits, has to be decided first. Usually, the system designers intend to give an
overall equivalent filter response equal to a raised-cosine filter response. From this point of view, this demo flow takes a raised-cosine filter response as an example.
Note that the raised-cosine filter should be also sketched in the sampling rate that matches the one of the primitive symbol stream, as shown in Fig. 4-16(c). As a result, the signals are convolved with the equivalent filter impulse response as presented in Fig. 4-16(d). Then the colored noise is added to the signals in the receiver side.
The signals before the ADC circuits are simulated in an analog-like waveforms. As a result, the system performance depends on the synchronization algorithm. Fig.
4-16(f)-(i) illustrate the down-sampled symbol stream using 1x symbol-rate down sampling, 2x and 4x symbol-rate down sampling. Moreover, the 1x symbol-rate down sampling also shows the sampling instance at the best and the worst sampling phases.
Accordingly, the simulation flow and methodology allow the platform to simulate different down-sampling rate behaviors and also the sampling phase offset phenomenon.
-1 1 3 5 7 9 11 13 15 17 19 21 23 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
Primitive symbol stream
Time index (nT)
Amplitude (unit strength)
(a)
Primitive Symbol Stream
-1 1 3 5 7 9 11 13 15 17 19 21 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
Primitive symbol stream in a continuous waveform
Time index (nT)
Amplitude (unit strength)
(b)
-8 -6 -4 -2 0 2 4 6 8
-0.2 0 0.2 0.4 0.6 0.8 1
Impulse response of a raised-cosine function
Time index (nT)
Amplitude
roll-off factor(β) = 0.2
(c)
Up-Sampled Symbol Stream
Filter Impulse Response
-7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 -9 -7 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
Symbol waveform after filtering
Time index (nT)
Amplitude (unit strength)
(d)
-7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 -9 -7 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
Symbol waveform with noise
Time index (nT)
Amplitude (unit strength)
(e)
Filtered Symbol Stream
Channel Out
-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
1x down-sampled symbol stream at the best samplings
Time index (nT)
Amplitude (unit strength)
(f)
-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
1x down-sampled symbol stream at the worst samplings
Time index (nT)
Amplitude (unit strength)
(g)
Down-Sampled Symbol Stream
(1x) (Best Sampling)
Down-Sampled Symbol Stream
(1x) (Worst Sampling)
-15-11 -7 -3 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
2x down-sampled symbol stream
Time index (nT)
Amplitude (unit strength)
(h)
-15 -11 -7 -3 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
4x down-sampled symbol stream
Time index (nT)
Amplitude (unit strength)
(i)
Down-Sampled Symbol Stream
(2x)
Down-Sampled Symbol Stream
(4x)
-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 -0.15
-0.1 -0.05 0 0.05 0.1 0.15
1x down-sampled symbol stream at the best samplings
Time index (nT)
Amplitude (unit strength)
(j)
Fig. 4-16. The flow and methodology for DSTC simulations
Fig. 4-17 plots the overall system performance. The solid curve, s(ε), represents the SNR required to reach a packet error rate (PER) of 8%, where whole packets are sampled at a fixed and identical sampling offset ε. When the DSTC algorithm is applied, the optimal sampling instance is sought during the preamble. Before the end of the preamble, the DSTC decides which timing instance is the best for sampling in terms of system performance. Since the DSTC is operated in a noisy environment, it does not always choose the best sampling instance. Consequently, the dotted-curve, p(ε), represents the probability of the final decision made by the DSTC. The system with sinc function has a stable performance over every possible sampling instance phase. Since a sinc function has a transfer function equivalent to the rectangular filter in frequency domain, the added noise power level to signal power over the whole signal bandwidth is the same. In other words, the noise is equally applied to the signals in terms of spectrum domain. Therefore, a sampling error causes system performance degradation in terms of two factors. One is the mentioned SIR feature, and another is the noise level. In this evaluation case, the decoding process can be
After Synchronization
achieved in the sub-5dB SNR level. The noise power in this level is much higher than the interference level. In addition, the noise is evenly applied to the signals, so the resulting system performances (required SNR at 8% PER) are almost the same with variant sampling timing error. Therefore, the SNR of the system required to reach PER=8% is given by
−
∫
= = ⋅ ⋅
5 . 0
5 . 0
%
8 s(ε) p(ε) dε
SNRPER (4-24)
It is also found that this probability function is correlated to the characteristic function derived in the proposed DSTC section. Although a sinc function condition does not provide clear characteristic to identify which instance is the best for signal sampling, the stable system performance has little impact due to the sampling error.
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 2.5
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Sampling Timing Error: ε Required SNR(dB) at PER=8%, s(ε)
Sinc function Required SNR at PER=8%
Probability of DSTC Timing Estimation
0 30
20
10
Probability of An Estimation (%)
Fig. 4-17. System performances on a corresponding sampling instance and the determined sampling-timing probability with DSTC algorithm in the sinc function condition
The same simulation conditions are applied with only the filter changed to a raised-cosine filter with roll-off factor 0.2~0.8. The resulting performance with variant sampling error is depicted in Figs. 4-18~4-20. The one with Butterworth filter is illustrated in Fig. 4-21. A larger roll-off factor implies a filter with less sharp frequency edge response. This causes the sideband signals more easily interfered by AWGN noise. In the time-domain expression, an inaccurate sampling instance will not only be abrupt by filter interference but also suffer from more noise power.
Consequently, the largest roll-off factor filter provides the most performance difference in the best and worst sampling instances as shown in Fig. 4-20. Moreover, the best sampling in this case suffers from the least noise, including the filter interference and AWGN noise. This implies that an incorrect sampling position has a large amount noise power added to the signal, although the overall signal power to the noise power ratio is the same in the scope of a packets. The probability to determine the proper sampling instances can be also found in the same figure with the dotted curve. As in the previous discussion on the characteristic function, the system with roll-off factor 0.8 provides the most obvious difference on the characteristic function, so the resulting probability curve as shown in Fig. 4-20 dotted curve is also correlated to the characteristic function. It is found in this case that the most probably calculated sampling phases provides the best system performance, i.e. the smallest SNR to reach 8% packet error rate. The case with a Butterworth filter has a performance promising the similar results, except the performance curve difference on different sampling instance and the calculated best sampling instance probability curve.
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5
Sampling Timing Error: ε Required SNR(dB) at PER=8%, s(ε)
Raised-cosine filter with roll-off factor 0.2 Required SNR at PER=8%
Probability of DSTC Timing Estimation
0 30
20
10
Probability of An Estimation (%)
Fig. 4-18. System performances on a corresponding sampling instance and the determined sampling-timing probability with DSTC algorithm in the RC function condition
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5
Sampling Timing Error: ε Required SNR(dB) at PER=8%, s(ε)
Raised-cosine filter with roll-off factor 0.5 Required SNR at PER=8%
Probability of DSTC Timing Estimation
0 30
20
10
Probability of An Estimation (%)
Fig. 4-19. System performances on a corresponding sampling instance and the determined sampling-timing probability with DSTC algorithm in the RC function condition
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5
Sampling Timing Error: ε Required SNR(dB) at PER=8%, s(ε)
Raised-cosine filter with roll-off factor 0.8 Required SNR at PER=8%
Probability of DSTC Timing Estimation
0 30
20
10
Probability of An Estimation (%)
Fig. 4-20. System performances on a corresponding sampling instance and the determined sampling-timing probability with DSTC algorithm in the RC function condition
-0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5
Sampling Timing Error: ε Required SNR(dB) at PER=8%, s(ε)
5th order Butterworth filter Required SNR at PER=8%
Probability of DSTC Timing Estimation
0 30
20
10
Probability of An Estimation (%)
Fig. 4-21. System performances on a corresponding sampling instance and the determined sampling-timing probability with DSTC algorithm in the Butterworth function condition
Figs. 4-22 and 4-23 shows an example of the decoded QPSK symbol constellation with the best and worst sampling instances, respectively. The evaluation
case applies a raised cosine filter with roll-off factor 0.5 at SNR 7dB for clearer performance recognition. The best-sampled constellation in Fig. 4-22 has much more condensed signal distribution than that in Fig. 4-23 from the worst signal samplings.
Fig. 4-22. The decoded QPSK symbol constellation with the best sampling instance in β =0.5 raised-cosine filter condition
Fig. 4-23. The decoded QPSK symbol constellation with the worst sampling instance in β =0.5 raised-cosine filter condition
The system performance is illustrated in Fig. 4-24 using a two-time interpolation scheme. The interpolation is achieved by taking twice samples with signals averaged, and is then proceeded by the data decoding. The sampling phase is simulated by 32 possible partitions. The simulated phase is fixed at a certain value as an initial phase.
Then the rest of the packet is decoded by the identical phase position. So, the overall performance is done by taking the initial phase pair of {ε/T, (ε/T+1/2T)} with ε=0, 1/32, 2/32, …, 15/32. The equivalent system filter is also assumed with the reference filters, i.e. sinc filter, raised-cosine filter, and Butterworth filter. The simulated system performance is shown in Fig. 4-24. Since the sampling phase over a packet is fixed and the selected initial phase is a random variable, it is reasonable to assume the targeted initial phase as a uniform distribution. Accordingly, each performance curve in Fig. 4-24 comes from the averaged performance with equally probably initial phase.
The system with sinc filter and low roll-off factor raised-cosine filter provide better packet error rate performance. This is due to the two-time sampled signals that can average and reduce the noise power level for a better signal quality decoding. On the contrary, the performance with the roll-off factor 0.8 raised-cosine filter requires 4.59dB SNR to reach the acceptable performance. The interpolation in this case does not benefit because the other additional sample signal may contribute unnecessary interference or noise on the original desired signals, and the additional interference or noise may have much more power level that can not be averaged even two or more samples are taken together. Note that this performance is the result from the expectation value. This implies that the performance in a specific sampling phase may result in a smaller SNR value than 4.59dB to reach the required performance, but this also implies that most of the cases require more than 4.59dB for the corresponding system performance. Although the resulted system curve is the outcome in average, it
reflects a fact that the system performance is not robust and not stable due to the variation in the performance.
-1 0 1 2 3 4 5 6 7
10-3 10-2 10-1 100
SNR (dB)
Packet error rate (PER)
System performance with 2x interpolation Sinc filter
Raised-cosine filter (β) = 0.2 Raised-cosine filter (β) = 0.5 Raised-cosine filter (β) = 0.8 Butterworth filter
Fig. 4-24. System performance with 2-time interpolation scheme
Table 4-1 summarizes the required SNR value to reach a 8% PER level in terms of the proposed DSTC and the 2x interpolation approach. If the equivalent system filter is an ideal sinc function, the required SNR with DSTC performs worse than that with interpolation. With reasonable system filter designs, say a 0.5 roll-off factor, the proposed DSTC has only about 0.69dB SNR loss with only half sampling frequency and a robust and stable system decoding performance. When a 0.8 roll-off factor is applied to the system design, the system with DSTC outperforms that with interpolation. The DSTC provides a stable decoding performance with 2.92dB SNR to reach 8% PER, but the interpolation only requires higher SNR values but also has poorer system stability. Another example with Butterworth has an about 0.3dB SNR loss for the tradeoff among sampling frequency, system stability, and system
performance. The discussion on the power consumption with the reduced sampling frequency may be further directed to chapter 6 for detailed illustration.
TABLE 4-1.SYSTEM PERFORMANCE SUMMARY WITH DSTC AND 2X INTERPOLATION Equivalent Filters SNR with DSTC (dB) SNR with 2x interpolation (dB)
Sinc 3.9251 1.93
Raised-cosine (β=0.2) 3.8948 1.89
Raised-cosine (β=0.5) 3.6166 2.9
Raised-cosine (β=0.8) 2.9244 4.59
Butterworth 3.3418 3.02
In hardware realization, there is finite resolution for controllable sampling timing.
So, a sub-optimum εsubopt solution of Eq. (4-23) can be expressed by
1 2
0~( 1) / 0
arg max s [ ]
opt
N
subopt s k l kT l
k K l n
kT r n
ε l = − −
= ∈ =
= =
∑
(4-25)
where N=128 in the case of MB-OFDM UWB. This means how close εopt and εsubopt
can be is limited by the hardware finite resolution. The value of variable l determines three important tradeoffs: (i) the performance loss when an incorrect εsubopt,err is made with a non-zero offset index.
, ˆ ( ) /
subopt err subopt Kopt offset T ls
ε =ε − =ε + (4-26)
where εˆ is the final decision; (ii) the PTCG hardware capability giving a lead or lag clock phase offset with resolution Ts