1. An Efficient Software Radio Implementation of the UMTS Turbo Codec
Matthew C. Valenti
Assistant Professor
Lane Dept. of Comp. Sci. & Elect. Eng.
West Virginia University
Morgantown, WV

2. UMTS Turbo Encoder From ETSI TS 125 212 v3.4.0 (2000-09)
Systematic
Output Xk
Input Xk
“Upper”
RSC
Encoder
Uninterleaved
Parity Zk
Output
“Lower”
RSC
Encoder
Interleaved
Parity
Z’k
Interleaver
Interleaved
Input
X’k
From ETSI TS v3.4.0 ( )
UMTS Multiplexing and channel coding
Data is segmented into blocks of L bits.
where 40  L  5114
© 2001

3. UMTS Interleaver: Inserting Data into Matrix
Data is fed row-wise into a R by C matrix.
R = 5, 10, or 20.
8  C  256
If L < RC then matrix is padded with dummy characters. X1 X2 X3 X4 X5 X6 X7 X8 X9
X10
X11
X12
X13
X14
X15
X16
X17
X18
X19
X20
X21
X22
X23
X24
X25
X26
X27
X28
X29
X30
X31
X32
X33
X34
X35
X36
X37
X38
X39
X40
© 2001

4. UMTS Interleaver: Intra-Row Permutations
Data is permuted within each row.
Permutation rules are rather complicated.
See spec for details. X2 X6 X5 X7 X3 X4 X1 X8
X10
X12
X11
X15
X13
X14 X9
X16
X18
X22
X21
X23
X19
X20
X17
X24
X26
X28
X27
X31
X29
X30
X25
X32
X40
X36
X35
X39
X37
X38
X33
X34
© 2001

5. UMTS Interleaver: Inter-Row Permutations
Rows are permuted.
If R = 5 or 10, the matrix is reflected about the middle row.
For R=20 the rule is more complicated and depends on L.
See spec for R=20 case.
X40
X36
X35
X39
X37
X38
X33
X34
X26
X28
X27
X31
X29
X30
X25
X32
X18
X22
X21
X23
X19
X20
X17
X24
X10
X12
X11
X15
X13
X14 X9
X16 X2 X6 X5 X7 X3 X4 X1 X8
© 2001

6. UMTS Interleaver: Reading Data From Matrix
Data is read from matrix column-wise.
Thus:
X’1 = X40 X’2 = X26 X’3 = X18 …
X’38 = X24 X’2 = X16 X’40 = X8
X40
X36
X35
X39
X37
X38
X33
X34
X26
X28
X27
X31
X29
X30
X25
X32
X18
X22
X21
X23
X19
X20
X17
X24
X10
X12
X11
X15
X13
X14 X9
X16 X2 X6 X5 X7 X3 X4 X1 X8
© 2001

7. Recursive Systematic Convolutional (RSC) Encoder
Systematic Output
(Upper Encoder Only)
Parity Output
(Both Encoders) D D D
Upper and lower encoders are identical:
Feedforward generator is 15 in octal.
Feedback generator is 13 in octal.
© 2001

8. Trellis Termination
XL+1 XL+2 XL+3
ZL+1 ZL+2 ZL+3 D D D
After the Lth input bit, a 3 bit tail is calculated.
The tail bit equals the fed back bit.
This guarantees that the registers get filled with zeros.
Each encoder has its own tail.
The tail bits and their parity bits are transmitted at the end.
© 2001

9. Total number of coded bits = 3L + 12
Output Stream Format
The format of the output steam is:
X1 Z1 Z’1 X2 Z2 Z’2 … XL ZL Z’L XL+1 ZL+1 XL+2 ZL+2 XL+3 ZL+3 X’L+1 Z’L+1 X’L+2 Z’L+2 X’L+3 Z’L+3
L data bits and
their associated
2L parity bits
(total of 3L bits)
3 tail bits for
upper encoder
and their
3 parity bits
3 tail bits for
lower encoder
and their
3 parity bits
Total number of coded bits = 3L + 12
Code rate:
© 2001

10. Channel Model and LLRs Channel gain: a Noise
Xk = {0,1}
Sk = {-1,1} Yk
R(Xk)
BPSK
Modulator ak nk
Channel gain: a
Rayleigh random variable if Rayleigh fading
a = 1 if AWGN channel
Noise
variance is:
© 2001

11. SISO-MAP Decoding Block
Decoder
V(Xk)
(Xk)
R(Zk)
Two input streams:
V(Xk): All information about systematic bit.
V(Xk) = R(Xk) + w(Xk)
w(Xk) is a priori information passed from other decoder.
Is the extrinsic information produced by other decoder.
R(Zk) observed parity bit for this constituent code.
One output stream:
Log-likelihood ratio:
© 2001

12. Turbo Decoding Architecture
w (Xk)
V1(Xk)
“Upper”
MAP
Decoder
R(Xk)
1(Xk)
R(Zk)
V2(Xk)
R(Z’k)
Interleave
V2(X’k)
“Lower”
MAP
Decoder
2(X’k)
2(Xk)
Deinnterleave
Initialization and timing:
w(Xk) is initialized to all zeros.
Upper decoder executes first, then lower decoder.

13. Log-MAP Algorithm: Overview
Log-MAP algorithm is MAP implemented in log-domain.
Multiplications become additions.
Additions become special “max*” operator (Jacobi logarithm)
Log-MAP is similar to the Viterbi algorithm.
Except “max” is replaced by “max*” in the ACS operation.
Processing:
Sweep through the trellis in forward direction using modified Viterbi algorithm.
Sweep through the trellis in backward direction using modified Viterbi algorithm.
Determine LLR for each trellis section.
Determine output extrinsic info for each trellis section.
© 2001

14. The max* operator max* must implement the following operation:
Ways to accomplish this:
C-function calls or large look-up-table.
Max operator.
Rough correction value.
Linear approximation.
log-MAP
max-log-MAP
constant-log-MAP
© 2001
linear-log-MAP

15. The Correction Function
linear-log-MAP
Constant-log-MAP
fc(|y-x|)
log-MAP
|y-x|

16. The Trellis for UMTS Dotted line = data 0 Solid line = data 1
Note that each node has one each of data 0 and 1 entering and leaving it.
The branch from node Si to Sj has metric ij
 00 S0 S0
 10 S1 S1 S2 S2 S3 S3 S4 S4 S5 S5 S6 S6
data bit associated
with branch Si Sj
parity bit associated
with branch Si Sj S7 S7
© 2001

17. Forward Recursion
A new metric must be calculated for each node in the trellis using:
where i1 and i2 are the two states connected to j.
Start from the beginning of the trellis (i.e. the left edge).
Initialize stage 0:
o = 0
i = - for all i  0
 00
’0
 0
 10
’1
 1
’2
 2
’3
 3
’4
 4
’5
 5
’6
 6
’7
 7
© 2001

18. Backward Recursion
A new metric must be calculated for each node in the trellis using:
where j1 and j2 are the two states connected to i.
Start from the end of the trellis (i.e. the right edge).
Initialize stage L+3:
o = 0
i = - for all i  0
 00
0
’0
 10
1
’1
2
’2
3
’3
4
’4
5
’5
6
’6
7
’7
© 2001

19. Log-likelihood Ratio The likelihood of any one branch is:
The likelihood of data 1 is found by summing the likelihoods of the solid branches.
The likelihood of data 0 is found by summing the likelihoods of the dashed branches.
The log likelihood ratio (LLR) is:
 00
 0
0
 10 1
1
 2
2
 3
3
 4
4 5
5 6
6
 7
7
© 2001

20. Memory Usage Issues No need to store both alpha and beta in trellis.
Compute beta first and store in trellis.
Then compute alpha and derive LLR estimates at the same time.
No need to store alpha trellis.
The metrics keep getting larger and larger.
Floating point: loss of precision.
Fixed point: overflow.
Solution: normalize the metrics:
An alternative normalization saves memory:
© 2001

21. Performance Comparison: 5114 bit UMTS turbo code10 -1 10
Fading -2 10 -3 10
AWGN
BER -4 10
constant-
log-MAP -5 10
log-MAP -6 10
linear-
log-MAP
max-log-MAP -7 10
0.5 1
1.5 2
Eb/No in dB

22. Average Number of Decoder Iterations16 14 12
constant-
log-MAP 10
log-MAP
avg. number of iterations 8
max-log-MAP 6 4
linear-log-MAP 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
Eb/No in dB

23. Dynamic Halting
Turbo decoding progresses until a fixed number of iterations have completed.
However, the decoder will often converge early.
Can stop once it has converged (i.e. BER = 0).
Stopping early can greatly increase the throughput.
For a simulation, it is reasonable to automatically halt once the BER goes to zero.
Requires knowledge of the data.
For an actual system, a “blind” method for halting is desirable.
Blind --- don’t know the data.
Simple blind estimator: Stop when LLR exceeds threshold.
© 2001

24. Conclusion Features of the decoder: For more information, visit:
Accurate and efficient max* calculation.
Simplified branch metrics.
Reduced storage by proper normalization.
Simple but effective blind halting technique.
For more information, visit:
A DOS demonstration of the code is available.
Also, these slides are available.
This work was sponsored by the Office of Naval Research and Mercury Computer Systems.
© 2001