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Course by: Tuan Do-Hong. E-mail the author

# Application of Viterbi Equalizer in GSM System

Module by: Sinh Nguyen-Le, Tuan Do-Hong. E-mail the authorsEdited By: Sinh Nguyen-Le, Tuan Do-HongTranslated By: Sinh Nguyen-Le, Tuan Do-Hong

The GSM time-division multiple access (TDMA) frame in Figure 1 has duration of 4.615 ms and comprising 8 slots, one assigned to each active mobile user. A normal transmission burst occupying one time slot contains 57 message bits on each side of a 26-bit midamble, called a training or sounding sequence. The slot-time duration is 0.577 ms (or the slot rate is 1733 slots/s). The purpose of the midamble is to assist the receiver in estimating the impulse response of the channel adaptively (during the time duration of each 0.577 ms slot). For the technique to be effective, the fading characteristics of the channel must not change appreciably during the time interval of one slot.

Consider a GSM receiver used aboard a high-speed train, traveling at a constant velocity of 200 km/hr (55.56 m/s). Assume the carrier frequency to be 900 MHz (the wavelength is λ = 0.33 m). The distance corresponding to a half-wavelength is traversed in T0λ/2V3T0λ/2V3 size 12{T rSub { size 8{0} } approx { { {λ} slash {2} } over {V} } approx 3} {} corresponds approximately to the coherence time. Therefore, the channel coherence time is more than five times greater than the slot time of 0.577 ms. The time needed for a significant change in channel fading characteristics is relatively long compared to the time duration of one slot.

The GSM symbol rate (or bit rate, since the modulation is binary) is 271 kilosymbols/s; the bandwidth, W, is 200 kHz. Since the typical rms delay spread στστ size 12{σ rSub { size 8{τ} } } {} in an urban environment is on the order of 2μs, then the resulting coherence bandwidth:

f 0 1 τ 100 kHz f 0 1 τ 100 kHz size 12{f rSub { size 8{0} } approx { {1} over {5σ rSub { size 8{τ} } } } approx "100"" kHz"} {}

Since f0<Wf0<W size 12{f rSub { size 8{0} } <W} {} , the GSM receiver must utilize some form of mitigation to combat frequency-selective distortion. To accomplish this goal, the Viterbi equalizer is typically implemented.

Figure 2 shows the basic functional blocks used in a GSM receiver for estimating the channel impulse response.

This estimate is used to provide the detector with channel-corrected reference waveforms as explained below: (the Viterbi algorithm is used in the final step to compute the MLSE of the message bits)

Let str(t)str(t) size 12{s rSub { size 8{ ital "tr"} } $$t$$ } {} be the transmitted midamble training sequence, and rtr(t)rtr(t) size 12{r rSub { size 8{ ital "tr"} } $$t$$ } {} be the corresponding received midamble training sequence. We have:

r tr ( t ) = s tr ( t ) h c ( t ) r tr ( t ) = s tr ( t ) h c ( t ) size 12{r rSub { size 8{ ital "tr"} } $$t$$ =s rSub { size 8{ ital "tr"} } $$t$$ * h rSub { size 8{c} } $$t$$ } {}

At the receiver, since rtr(t)rtr(t) size 12{r rSub { size 8{ ital "tr"} } $$t$$ } {} is part of the received normal burst, it is extracted and sent to a filter having impulse response hmf(t)hmf(t) size 12{h rSub { size 8{ ital "mf"} } $$t$$ } {} , that is matched to str(t)str(t) size 12{s rSub { size 8{ ital "tr"} } $$t$$ } {}. This matched filter yields at its output an estimate of hc(t)hc(t) size 12{h rSub { size 8{c} } $$t$$ } {}, denoted he(t)he(t) size 12{h rSub { size 8{e} } $$t$$ } {}:

h e ( t ) = r tr ( t ) h mf ( t ) h e ( t ) = r tr ( t ) h mf ( t ) size 12{h rSub { size 8{e} } $$t$$ =r rSub { size 8{ ital "tr"} } $$t$$ * h rSub { size 8{ ital "mf"} } $$t$$ } {}

= s tr ( t ) h c ( t ) h mf ( t ) = s tr ( t ) h c ( t ) h mf ( t ) size 12{ {}=s rSub { size 8{ ital "tr"} } $$t$$ * h rSub { size 8{c} } $$t$$ * h rSub { size 8{ ital "mf"} } $$t$$ } {}

where Rs(t)=str(t)hmf(t)Rs(t)=str(t)hmf(t) size 12{R rSub { size 8{s} } $$t$$ =s rSub { size 8{ ital "tr"} } $$t$$ * h rSub { size 8{ ital "mf"} } $$t$$ } {} is the autocorrelation function of str(t)str(t) size 12{s rSub { size 8{ ital "tr"} } $$t$$ } {}. If str(t)str(t) size 12{s rSub { size 8{ ital "tr"} } $$t$$ } {} is designed to have a highly-peaked (impulse-like) autocorrelation function Rs(t)Rs(t) size 12{R rSub { size 8{s} } $$t$$ } {}, then he(t)hc(t)he(t)hc(t) size 12{h rSub { size 8{e} } $$t$$ approx h rSub { size 8{c} } $$t$$ } {}.

Next, we use a windowing function, w(t)w(t) size 12{w $$t$$ } {}, to truncate he(t)he(t) size 12{h rSub { size 8{e} } $$t$$ } {} to form a computationally affordable function, hw(t)hw(t) size 12{h rSub { size 8{w} } $$t$$ } {}. The time duration of w(t)w(t) size 12{w $$t$$ } {}, denoted L0L0 size 12{L rSub { size 8{0} } } {}, must be large enough to compensate for the effect of typical channel-induced ISI. The term L0L0 size 12{L rSub { size 8{0} } } {} consists of the sum of two contributions, namely LCISILCISI size 12{L rSub { size 8{ ital "CISI"} } } {}, corresponding to the controlled ISI caused by Gaussian filtering of the baseband waveform (which then modulates the carrier using MSK), and LCLC size 12{L rSub { size 8{C} } } {}, corresponding to the channel-induced ISI caused by multipath propagation. Thus,

L 0 = L CISI + L C L 0 = L CISI + L C size 12{L rSub { size 8{0} } =L rSub { size 8{ ital "CISI"} } +L rSub { size 8{C} } } {}

The GSM system is required to provide distortion mitigation caused by signal dispersion having delay spreads of approximately 15–20 μs. Since in GSM the bit duration is 3.69 μs, we can express L0L0 size 12{L rSub { size 8{0} } } {} in units of bit intervals. Thus, the Viterbi equalizer used in GSM has a memory of 4–6 bit intervals. For each L0L0 size 12{L rSub { size 8{0} } } {}-bit interval in the message, the function of the Viterbi equalizer is to find the most likely L0L0 size 12{L rSub { size 8{0} } } {}-bit sequence out of the 2L02L0 size 12{2 rSup { size 8{L rSub { size 6{0} } } } } {} possible sequences that might have been transmitted.

Determining the most likely transmitted L0L0 size 12{L rSub { size 8{0} } } {}-bit sequence requires that 2L02L0 size 12{2 rSup { size 8{L rSub { size 6{0} } } } } {} meaningful reference waveforms be created by disturbing) the 2L02L0 size 12{2 rSup { size 8{L rSub { size 6{0} } } } } {} ideal waveforms (generated at the receiver) in the same way that the channel has disturbed the transmitted slot. Therefore, the 2L02L0 size 12{2 rSup { size 8{L rSub { size 6{0} } } } } {} reference waveforms are convolved with the windowed estimate of the channel impulse response, hw(t)hw(t) size 12{h rSub { size 8{w} } $$t$$ } {} in order to generate the disturbed or so-called channel-corrected reference waveforms.

Next, the channel-corrected reference waveforms are compared against the received data waveforms to yield metric calculations. However, before the comparison takes place, the received data waveforms are convolved with the known windowed autocorrelation function w(t)Rs(t)w(t)Rs(t) size 12{w $$t$$ R rSub { size 8{s} } $$t$$ } {}, transforming them in a manner comparable to the transformation applied to the reference waveforms. This filtered message signal is compared to all possible 2L02L0 size 12{2 rSup { size 8{L rSub { size 6{0} } } } } {} channel-corrected reference signals, and metrics are computed in a manner similar to that used in the Viterbi decoding algorithm. It yields the maximum likelihood estimate of the transmitted data sequence.

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