Exercise 3.4Z: GSM Full-Rate Voice Codec

From LNTwww

LPC, LTP and RPE parameters in the GSM Full-Rate Vocoder

This codec called "GSM Full-Rate Vocoder"  (which was standardized for the GSM system in 1991)  stands for a joint realization of coder and decoder and combines three methods for the compression of speech signals:

  • Linear Predictive Coding  $\rm (LPC)$,
  • Long Term Prediction  $\rm (LTP)$, and
  • Regular Pulse Excitation  $\rm (RPE)$.

The numbers shown in the graphic indicate the number of bits generated by the three units of this Full-Rate speech codec per frame of  $20$  millisecond duration each.

It should be noted that LTP and RPE, unlike LPC, do not work frame by frame, but with sub-blocks of  $5$  milliseconds.  However, this has no influence on solving the task.

The input signal in the above graphic is the digitalized speech signal  $s_{\rm R}(n)$.

This results from the analog speech signal  $s(t)$  by

  • a suitable limitation to the bandwidth $B$,
  • sampling at the sampling rate $f_{\rm A} = 8 \ \rm kHz$,
  • quantization with $13 \ \rm bit$,
  • following segmentation into blocks of each $20 \ \rm ms$.

The further tasks of preprocessing will not be discussed in detail here.




To which bandwidth $B$  must the speech signal be limited?

$B \ = \ $

$\ \rm kHz$


Of how many samples  $(N_{\rm R})$  is there a speech frame?  How large is the input data rate $R_{\rm In}$?

$N_{\rm R} \hspace{0.18cm} = \ $

$\ \rm samples$
$R_{\rm In} \hspace{0.15cm} = \ $

$\ \rm kbit/s$


What is the output data rate $R_{\rm Out}$ of the GSM–full rate codec?

$R_{\rm Out} \ = \ $

$\ \rm kbit/s$


Which statements apply to the block "LPC"?

LPC makes a short-term prediction over one millisecond.
The  $36$  LPC bits specify coefficients that the receiver uses to undo the LPC filtering.
The filter for short-term prediction is recursive.
The LPC output signal is identical to the input signal  $s_{\rm R}(t)$.


Which statements regarding the block "LTP" are true?

LTP removes periodic structures of the speech signal.
The long-term prediction is performed once per frame.
The memory of the LTP predictor is up to  $15 \ \rm ms$.


Which statements apply to the block "RPE"?

RPE delivers fewer bits than LPC and LTP.
RPE removes unimportant parts for the subjective impression.
RPE subdivides each sub-block into four sub-sequences.
RPE selects the sub-sequence with the minimum energy.


(1)  To satisfy the sampling theorem, the bandwidth $B$  must not exceed  $ f_{\rm A}/2 \hspace{0.15cm}\underline{= 4 \ \ \rm kHz}$.

(2)  The given sampling rate  $f_{\rm A} = 8 \ \rm kHz$  results in a distance between individual samples of  $T_{\rm A} = 0.125 \ \rm ms$.

  • Thus a speech frame of  $20 {\rm ms}$  consists of  $N_{\rm R} = 20/0.125 = \underline{160 \ \rm samples}$, each quantized with  $13 \ \rm bit$.
  • The data rate is thus
$$R_{\rm In} = \frac{160 \cdot 13}{20 \,{\rm ms}} \hspace{0.15cm} \underline {= 104\,{\rm kbit/s}}\hspace{0.05cm}.$$

(3)  The graph shows that per speech frame  $36 \ {\rm (LPC)} + 36 \ {\rm (LTP)} + 188 \ {\rm (RPE)} = 260 \ \ \rm bit$  are output.

  • From this the output data rate is calculated as
$$R_{\rm Out} = \frac{260}{20 \,{\rm ms}} \hspace{0.15cm} \underline {= 13\,{\rm kbit/s}}\hspace{0.05cm}.$$
  • The compression factor achieved by the full-rate speech codec is thus  $104/13 = 8$.

(4)  The first two statements are true:

  • The 36 LPC bits describe a total of eight filter coefficients of a non-recursive filter, whereby eight  $\rm ACF$ values are determined from the short-term analysis and where these are converted into reflection factors  $r_{k}$  after the so-called "Schur recursion".
  • From these the eight LAR coefficients are calculated according to the function  ${\rm ln}\big[(1 - r_{k})/(1 + r_{k})\big]$, quantized with a different number of bits and sent to the receiver.
  • The LPC output signal has a significantly lower amplitude than its input  $s_{\rm R}(n)$, and it has a significantly reduced dynamic range and a flatter spectrum.

(5)  Correct are the statements 1 and 3, but not the second:

  • The LTP analysis and filtering is done blockwise every  $5 \ \rm ms$  ⇒   $(40$  samples$)$, i.e. four times per speech frame.
  • The cross correlation function  $\rm (CCF)$  between the current sub-block and the three previous sub-blocks is formed.
  • For each sub-block, an LTP delay and an LTP gain are determined which best match the sub-block.
  • A correction signal of the following component "RPE" is also taken into account.
  • For the long-term prediction, as with the LPC, the output is reduced in redundancy compared to the input.

(6)  The statements 2 and 3 are correct:

  • The fact that statement 1 is wrong can be seen from the graphic on the data page, because  $188$  of the  $260$  output bits come from the RPE.  Voice would be understandable with RPE alone (without LPC and LTP).
  • Regarding the last statement:  The RPE is of course looking for the subsequence with the maximum energy.  The RPE pulses are a subsequence  $(13$  of  $40$  samples$)$  of three bits per subframe of  $5 \ \rm ms$  and accordingly  $12$  bits per  $20 \ \rm ms$  frame.
  • The "RPE pulse" thus occupies  $13 \cdot 12 = 156$  of the  $260$  output bits.

More details about the RPE block can be found on the page  RPE coding  of the book  "Examples of Communication Systems".