Low Threshold Telemetry with Noncausal Smoothing
Detailed Technical Review

Responsible Engineer:	Chau Buu
Presenter:	Jon Hamkins

Contents

• Description
• Functional Requirements
• Design
  • Theoretical basis for noncausal smoothing
  • Practical considerations
  • Block V Receiver considerations
  • Software simulation of Block V Receiver tracking loops
    • User-definable parameters
    • Program flow
• Results
  • $L(1) = 0 \mbox{ dBrad$^2$/Hz}$
  • $L(1) = -10 \mbox{ dBrad$^2$/Hz}$
  • $L(1) = -20 \mbox{ dBrad$^2$/Hz}$
  • $L(1) = -30 \mbox{ dBrad$^2$/Hz}$
  • $L(1) = -40 \mbox{ dBrad$^2$/Hz}$

Description

• Low-Threshold Telemetry is designed for low-data rate missions with weak signals and poor spacecraft oscillator stability.
  • Tracking loop bandwidth is limited by the carrier phase noise on the downlink
  • Reducing carrier phase error will enable tracking at lower carrier threshold
• Noncausal filter is used to minimize carrier phase tracking error in the presence of carrier phase noise, for an improvement of 5-7 dB in carrier tracking threshold over causal filters such as phase-locked loop.
• Carrier tracking errors are removed from the symbols prior to output to telemetry processor for improvement in bit SNR

Functional Requirements

From the TMOD Telemetry Service System Requirements Document (DTT1298), the receiver shall have no more than a 0.5 dB loss of symbol SNR due to carrier tracking, for the following parameters:

Symbol rate		4 to 490 symbols/second
Phase noise type		Frequency flicker dominated (i.e., PSD $\sim 1/f^3$)
Phase noise level		$-32 < L(1) < 0$ dBrad$^2$/Hz
$P_c/N_0$		$22 + 0.5 L(1)$ or greater

Functional Requirements Clarification

Old New

$\parbox{2in}{The system shall be able to track a residual carrier with a mean squared error of 0.1 square radians or less.}$ $\rightarrow$ $\parbox{2in}{The system shall provide symbol outputs whose SNR loss due to carrier tracking shall be no more than 0.5 dB.}$

For low data rates, the symbol SNR loss is governed by the average value of $\cos^2\phi_c$.

$$E[\cos^2\phi_c] = E[(1-\phi_c^2/2+O(\phi_c^4))^2] \approx 1 - E[\phi_c^2] \ge 0.9 = -0.46 \mbox{dB}$$

Thus, the requirement is virtually the same.

Design: Theoretical basis for noncausal smoothing

For random processes related by

according Wiener theory, the LMMSE of $X$ given $Y$ is given by $\hat X$


where

$$H(f) = \frac{S_X(f)}{S_X(f)+S_N(f)}$$

and where $S_X(f)$ and $S_N(f)$ are the 2-sided spectral densities.

Design: Theoretical basis for noncausal smoothing (cont.)

$$H(f) = \frac{S_X(f)}{S_X(f)+S_N(f)}$$

Since $H(f)$ is real and even, so is $h(t)$.
Therefore,

The LMMSE filter is noncausal.

When $S_X(f) = S_3/(2\vert f\vert^3)$ and $S_N(f) = N_0/2$ are the 2-sided PSD's, then

$$H(f) = \frac{1}{1+\vert f\vert^3N_0/S_3}.$$

Design: Practical considerations- using triangular filter shape

The carrier phase MSE for a filter is given by:

$$\sigma^2 = \underbrace{{BN_0 \over P_c}}_{\mbox{due to AWGN}} + \underbrace{{kS_3\over B^2}}_{\mbox{due to phase noise}}$$

$B$ is the filter bandwidth and $k$ depends only on the filter shape. The bandwidth minimizing $\sigma^2$ is

$$B = (2S_3kP_c/N_0)^{1/3},$$

resulting in an MSE of

$$\sigma^2 = {3\over 2}\left(2kS_3N_0^2\over P_c^2\right)^{1/3}.$$

We replace LMMSE filter with a triangle. We no longer have the LMMSE, but it's very close ($k=0.272$ vs. $k=0.262$, compared to $k=8.7$ for 2nd order PLL [#!Ki96!#]), and easy to implement. Its length is

$$N = \frac{2}{3BT_c} = \frac{0.816}{(S_3P_c/N_0)^{1/3}T_c}.$$

Notes:

• $N\rightarrow\infty$ as either $S_3\rightarrow 0$ or $P_c/N_0\rightarrow 0$
• $\sigma^2 \rightarrow 0$ as $S_3\rightarrow 0$ or $P_c/N_0\rightarrow \infty$.

Block diagram of the smoother filter

Block diagram of the Block V Receiver without smoother.

Block diagram of the Block V Receiver with smoother.

$\parbox{3in}{ Main signal points on BVR: \begin{eqnarray*} r[n] & = ... ... T_{HBF} + \hat\theta_{sc}[n]) \delta(\hat\theta_{sc}[n])\\ \end{eqnarray*}}$$\parbox{4in}{ \vspace*{-0.2in} \begin{eqnarray*} \mbox{where} \\ P_... ...heta_{sc}(t) - \hat\theta_{sc}(t) = \mbox{\small Subc. error} \end{eqnarray*}}$

Software Design: Signal Model Parameters

Signal type:	Residual carrier BPSK with data subcarrier
$P_t/N_0$:	23.01 dB
$P_c/N_0$:	20.00 dB
$P_d/N_0$:	20.00 dB
Symbol rate:	500 sym/sec
Modulation index:	45 deg (0.79 rad)
Carrier
Phase noise type:	Frequency-flicker noise
PSD (2-sided):	$0.01/f^3$ rad$^2$/Hz
Power at 1 Hz offset:	-20.00 dBrad$^2$/Hz
Phase stored modulo 2 pi:	No
Length of phase filter:	4096
Phase at t=0:	0 deg (0.00 rad)
Subcarrier
Shape:	Square wave
Phase stored modulo 2 pi:	No
Frequency:	22.5 kHz
Phase at t=0:	0 deg (0.00 rad)
Channel type:	AWGN, with carrier phase noise
Sample rate at ADC output:	160000000 Hz
Sample rate after HBF:	80000000 Hz

Software Design: Carrier Recovery Parameters

Carrier update rate:	2000 Hz
Causal
Loop type:	Standard underdamped PLL
Order:	2
PLL loop bandwidth:	3.26 Hz
Predicted tracking performance
MSE:	0.0326 (AWGN) + 0.0163 (phase noise) = 0.0490 rad$^2$
Effective loop SNR:	13.10 dB
Symbol SNR loss due to carrier tracking:	0.0104 dB
Noncausal
Loop Type:	Linear smoothing filter
Filter shape:	Triangular
Filter bandwidth:	1.03 Hz
Filter length:	1297
Frequency tracking method:	PLL-derived frequencies
Predicted performance
MSE:	0.0103 (AWGN) + 0.0051 (phase noise) = 0.0245 rad$^2$
Effective loop SNR:	18.12 dB
Gain over causal tracking:	5.02 dB
Symbol SNR loss due to carrier tracking:	0.0010 dB

Software Design: Subcarrier and Timing Recovery Parameters

Subarrier Recovery
Subcarrier update rate:	500 Hz
Loop type:	Standard underdamped PLL
Order:	2
PLL loop bandwidth:	0.100 Hz
Window fraction:	0.25
Predicted tracking performance
MSE:	0.0022 rad$^2$
Squaring loss:	5.44 dB
Loss due to carrier tracking:	0.11 dB
Effective loop SNR:	26.55 dB
Timing Recovery
Timing update rate:	500 Hz
Loop type:	Perfect Recovery assumed
Available Debugging Cheats:
Send all zeroes
AWGN turned off
Carrier phase noise turned off
Perfect carrier recovery
Smoother turned off

Design: Program flow

main(int argc, char *argv[])
{
   process_command_line_arguments(argc,argv); /* see args.c                 */
   initialize_global_variables(); /*  e.g., bandwidths, memory allocation   */
   print_header();                /* info about simulation parameters       */
   for(k=0; ck<number_of_updates; k++) {
      get_ADC_output_sample();    /* output of analog to digital convertor  */
      get_NCO_mixer_output();     /* output after NCO mixer                 */
      get_HBF_output();           /* output of half-band filter             */
      get_subcarrier_correction();/* output after subcarrier mixer          */
      update_accumulators();      /* for car., subc., timing, & smoother    */
      if (carrier_sample_is_ready())    process_carrier_sample();
      if (subcarrier_sample_is_ready()) process_subcarrier_sample();
      if (timing_sample_is_ready())     process_timing_sample();
      if (smoother_sample_is_ready())   process_smoother_sample();
      print_results();            /* print results every so often           */
   }
}

Design: Phase noise generation

• The input $X$ has 2-sided PSD $S_{XX} = 1$.
• We set $H(f) = \sqrt{S_3/(2\vert f\vert^3)}$, so that the 2-sided PSD of $Y$ is
  
  $$S_{YY}(f) = \vert H(f)\vert^2 S_{XX}(f) = S_3/(2\vert f\vert^3).$$
  
  
  (The 1-sided PSD is $S_3/f^3$.)
• Phase noise samples may be generated on an as-needed basis during a simulation.
• The convolution is implemented quickly with FFT's, and the overlap-add method.

Design: Phase noise generation

Measured PSD's of simulated phase noise

Results: $L(1) = 0 \mbox{ dBrad$^2$/Hz}$

Results: $L(1) = -10 \mbox{ dBrad$^2$/Hz}$

Results: $L(1) = -20 \mbox{ dBrad$^2$/Hz}$

Results: $L(1) = -30 \mbox{ dBrad$^2$/Hz}$

Results: $L(1) = -40 \mbox{ dBrad$^2$/Hz}$