JSC-CR-06-072 

DEPARTMENT OF DEFENSE 

JOINT SPECTRUM CENTER 

ANNAPOLIS, MARYLAND 21402-5064 

COMMUNICATIONS RECEIVER PERFORMANCE 

DEGRADATION HANDBOOK 

Prepared for 

National Telecommunications and Information Administration (NTIA) 

1401 Constitution Ave, NW 

Washington, DC 20230 

JSC Project Engineer 

Steven Molina 

NOVEMBER 2006 

CONSULTING REPORT

Prepared by 

Dr. Donald Wheeler, Nicholas Canzona 

Alion Science and Technology 

Under Contract to 

Department of Defense 

Approved for public release; distribution is unlimited.

REPORT DOCUMENTATION PAGE 

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1. REPORT DATE (DD-MM-YYYY)

11-2006 

2. REPORT TYPE

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3. DATES COVERED (From - To)

5a. CONTRACT NUMBER 

DCA100-00-C-4012

5b. GRANT NUMBER 

4. TITLE AND SUBTITLE 

Communications Receiver Performance Degradation Handbook 

5c. PROGRAM ELEMENT NUMBER 

5d. PROJECT NUMBER 

P1314

5e. TASK NUMBER 

6. AUTHOR(S) 

Wheeler, Dr. Donald; Canzona, Nicholas 

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT

NUMBER Joint Spectrum Center

RD&A Division (JSC/J5)

2004 Turbot Landing 

Annapolis, MD 21402-5064

JSC-CR-06-072 

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

NTIA 

11. SPONSOR/MONITOR’S REPORT 

NUMBER(S)

National Telecommunications and Information Administration 

1401 Constitution Ave, NW 

Washington, DC 20230 

12. DISTRIBUTION / AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACT

This handbook provides the radio frequency (RF) analyst with the capability to calculate the effects of noise and 

interference on RF communications receivers. A receiver is modeled as a sequence of modules. Each module has a 

transfer function that relates the module outputs to the module inputs. By consecutively analyzing each module in the 

sequence, the analyst can then relate the receiver outputs (performance) to the receiver inputs (signal characteristics). 

15. SUBJECT TERMS

communications receiver, frequency-dependent rejection, FDR, spread spectrum, demodulator, forward error 

correction, FEC, bit error rate, BER, source decoder, noise, interference 

16. SECURITY CLASSIFICATION OF: 

a. REPORT b. ABSTRACT c. THIS PAGE 

17. LIMITATION 

OF ABSTRACT

18. NUMBER 

OF PAGES

19a. NAME OF RESPONSIBLE PERSON

Steven Molina 

U U U SAR 161 19b. TELEPHONE NUMBER (include area code)

(410) 293-9261, DSN 281-9261 

Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std. Z39.18
 JSC-CR-06-072 

EXECUTIVE SUMMARY 

In May 2003, President Bush established the Spectrum Policy Initiative to promote the development and 

implementation of a United States spectrum policy for the 21st century. In response to the Spectrum

Policy Initiative, the Secretary of Commerce established a Federal Government Spectrum Task Force 

and initiated a series of public meetings to address policies affecting spectrum use by the federal 

government, state, and local governments, and the private sector. The recommendations resulting from

these activities were included in two reports released by the Secretary of Commerce in June 2004. In 

November 2004, the President directed the federal agencies to develop a plan to implement the 24 

recommendations contained in the reports. One of the recommendations directed the National 

Telecommunications and Information Administration (NTIA) to develop a handbook documenting best 

practices in spectrum engineering that will be recognized by all regulatory authorities in the United 

States. The Best Practices Handbook will address the electromagnetic compatibility (EMC) analysis 

process. Establishing receiver performance degradation levels is required to perform an EMC analysis. 

The NTIA requested that the Joint Spectrum Center develop a communications receiver performance 

degradation handbook that could be included as part of the Best Practices Handbook. 

The purpose of this handbook is to provide the radio frequency (RF) analyst with the capability to 

calculate the effects of noise and interference on RF communications receivers. A receiver is modeled 

as a sequence of modules. Each module has a transfer function that relates the module outputs to the 

module inputs. By consecutively analyzing each module in the sequence, the analyst can then relate the 

receiver outputs (performance) to the receiver inputs (signal characteristics). 

Section 2 describes the general procedures for performing a communications receiver performance 

analysis. It introduces the fundamental concepts and describes the receiver model. Sections 3-8 present 

the transfer functions for the receiver modules. Section 9 presents two examples that demonstrate how 

this handbook can be used to perform a receiver analysis. 

The first module in the receiver is designed to amplify the desired signal, convert it to an intermediate 

frequency (IF), and filter out some interference and noise. Section 3 provides a detailed description of

the RF/IF section. It also specifies how interfering signals may be changed by the filters. 

Section 4 describes the despreader module in spread spectrum receivers. It specifies how the spread 

spectrum processing gain can be calculated. It also provides information that can be used to analyze 

spread spectrum multiple-access systems. 

i 
JSC-CR-06-072 

Sections 5 and 6 describe the demodulator module. They provide plots that characterize performance as 

a function of the input power ratios. Analog voice and broadcast television receivers are considered in 

Section 5 and digital receivers are considered in Section 6. 

Section 7 describes forward error correction (FEC) decoders, which use redundancy bits to reduce the 

bit-error rate (BER) by correcting some of the bit errors introduced by interference and noise. It 

provides plots of output BER vs. input BER for several types of FEC decoders. 

Section 8 describes source decoders, which convert the information bit sequence to the final format at 

the receiver output. For a digital voice system, this format is an analog voice waveform. Section 8 

provides plots that relate output signal quality to the input BER. 

ii 
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TABLE OF CONTENTS 

GLOSSARY .............................................................................................................................................. ix 

SECTION 1 – INTRODUCTION ........................................................................................................... 1-1 

1.1 BACKGROUND .......................................................................................................................... 1-1 

1.2 PURPOSE OF THE HANDBOOK.............................................................................................. 1-1 

1.3 OVERVIEW OF HANDBOOK CONTENTS ............................................................................. 1-2 

1.4 HOW TO USE THE HANDBOOK ............................................................................................. 1-2 

SECTION 2 – ANALYSIS PROCEDURES........................................................................................... 2-1 

2.1 SYSTEM TYPES ......................................................................................................................... 2-1 

2.1.1 Communications.................................................................................................................... 2-1 

2.1.2 Non-communications.............................................................................................................2-2 

2.2 SIGNAL TYPES........................................................................................................................... 2-2 

2.2.1 Noise ...................................................................................................................................... 2-2 

2.2.2 Continuous-Wave ..................................................................................................................2-2 

2.2.3 Modulated .............................................................................................................................. 2-3 

2.2.4 Intermittent............................................................................................................................. 2-3 

2.3 LINK BUDGET ANALYSIS....................................................................................................... 2-3

2.3.1 Link Budget Model................................................................................................................2-3 

2.3.2 Link Budget Equation ............................................................................................................2-5 

2.3.3 Link Budget Performance Measures...................................................................................... 2-6 

2.3.4 Limitations of Link Budget Analysis.....................................................................................2-6 

2.4 RECEIVER PERFORMANCE ANALYSIS................................................................................ 2-7 

2.4.1 System Performance Model................................................................................................... 2-7 

2.4.2 System Performance Measures............................................................................................2-10 

2.4.3 System Performance Analysis Limitations.......................................................................... 2-11 

2.4.4 Receiver Performance Analysis Types................................................................................ 2-13 

2.4.5 Receiver Performance Analysis Steps................................................................................. 2-14

SECTION 3 – RF/IF SECTION ..............................................................................................................3-1 

3.1 DESCRIPTION ............................................................................................................................ 3-1 

3.2 FDR............................................................................................................................................... 3-2 

3.3 CHANGES TO INTERFERENCE WAVEFORM ......................................................................3-5 

3.3.1 Frequency Hoppers................................................................................................................3-5 

3.3.2 Pulsed Interfering Signals......................................................................................................3-6 

3.3.3 Digital Interfering Signals......................................................................................................3-9 

3.4 ANALYSIS WITH INTERMITTENT SIGNALS.....................................................................3-10 

3.5 SUMMARY................................................................................................................................ 3-12 

SECTION 4 – DESPREADER................................................................................................................4-1 

4.1 DESCRIPTION ............................................................................................................................ 4-1 

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4.2 DIRECT SEQUENCE..................................................................................................................4-2 

4.2.1 Processing Gain ..................................................................................................................... 4-3 

4.2.2 Multiple-Access Interference.................................................................................................4-4 

4.3 FREQUENCY HOPPING ............................................................................................................ 4-5 

4.3.1 Processing Gain ..................................................................................................................... 4-5 

4.3.2 Multiple-Access Interference.................................................................................................4-6 

4.4 TIME HOPPING .......................................................................................................................... 4-7 

4.5 SUMMARY.................................................................................................................................. 4-7 

SECTION 5 – ANALOG DEMODULATOR.........................................................................................5-1 

5.1 INTRODUCTION ........................................................................................................................ 5-1 

5.2 VOICE PERFORMANCE MEASURES .....................................................................................5-1 

5.2.1 Articulation Index .................................................................................................................. 5-2 

5.2.2 Articulation Score .................................................................................................................. 5-2 

5.3 AM VOICE................................................................................................................................... 5-4 

5.3.1 Description............................................................................................................................. 5-4 

5.3.2 Performance ........................................................................................................................... 5-4 

5.3.3 Transfer Function................................................................................................................... 5-5 

5.3.4 AI Curves............................................................................................................................... 5-6 

5.4 SINGLE SIDEBAND VOICE...................................................................................................... 5-9

5.4.1 Description............................................................................................................................. 5-9 

5.4.2 Transfer Function................................................................................................................... 5-9 

5.4.3 AI Curves............................................................................................................................. 5-10 

5.5 FM VOICE ................................................................................................................................. 5-13 

5.5.1 Description........................................................................................................................... 5-13 

5.5.2 Performance ......................................................................................................................... 5-14 

5.5.3 Transfer Function................................................................................................................. 5-15 

5.5.4 AI Curves............................................................................................................................. 5-16 

5.6 BROADCAST TELEVISION....................................................................................................5-19 

5.6.1 Description........................................................................................................................... 5-19 

5.6.2 TASO Curves....................................................................................................................... 5-19 

SECTION 6 – DIGITAL DEMODULATOR..........................................................................................6-1 

6.1 INTRODUCTION ........................................................................................................................ 6-1 

6.2 BINARY SYSTEMS ....................................................................................................................6-2 

6.3 M-ARY SYSTEMS ......................................................................................................................6-4 

6.4 M-ARY SYSTEM PERFORMANCE IN BINARY TERMS......................................................6-5 

6.5 TYPES OF INTERFERENCE...................................................................................................... 6-7

6.6 PSK ............................................................................................................................................... 6-8 

6.6.1 Description............................................................................................................................. 6-8 

6.6.2 BER Curves ......................................................................................................................... 6-10 

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6.7 DPSK .......................................................................................................................................... 6-15 

6.7.1 Description........................................................................................................................... 6-15 

6.7.2 BER Curves ......................................................................................................................... 6-15 

6.8 FSK ............................................................................................................................................. 6-16 

6.8.1 Description........................................................................................................................... 6-16 

6.8.2 BER Curves ......................................................................................................................... 6-20 

6.9 MINIMUM-SHIFT KEYING.....................................................................................................6-25 

6.9.1 Description........................................................................................................................... 6-25 

6.9.2 BER Curves ......................................................................................................................... 6-26 

6.10 QUASI-BANDLIMITED MSK ...............................................................................................6-26 

6.10.1 Description......................................................................................................................... 6-26 

6.10.2 BER Curves ....................................................................................................................... 6-29 

6.11 ASK .......................................................................................................................................... 6-29 

6.11.1 Description......................................................................................................................... 6-29 

6.11.2 BER Curves ....................................................................................................................... 6-31 

6.12 QUADRATURE AMPLITUDE MODULATION...................................................................6-37 

6.12.1 Description......................................................................................................................... 6-37 

6.12.2 BER Curves ....................................................................................................................... 6-38 

SECTION 7 – FEC DECODER ..............................................................................................................7-1 

7.1 INTRODUCTION ........................................................................................................................ 7-1 

7.2 HARD DECISION VS. SOFT DECISION.................................................................................. 7-1 

7.3 DECODER PERFORMANCE.....................................................................................................7-2 

7.4 TEMPORAL FLUCTUATIONS.................................................................................................. 7-2

7.5 BLOCK CODES........................................................................................................................... 7-3 

7.5.1 General Concepts and Terminology ...................................................................................... 7-3 

7.5.2 BER Calculation for Block Coding FEC...............................................................................7-5 

7.5.3 Bose-Chaudhurie-Hocquenghem Codes................................................................................ 7-6 

7.5.4 Hamming Code ...................................................................................................................... 7-9 

7.5.5 Hadamard Code ..................................................................................................................... 7-9 

7.5.6 Golay Code .......................................................................................................................... 7-12 

7.5.7 Reed-Solomon Codes...........................................................................................................7-12 

7.5.8 Interleaving .......................................................................................................................... 7-17 

7.6 CONVOLUTIONAL CODES....................................................................................................7-20 

7.6.1 Description........................................................................................................................... 7-20 

7.6.2 Decoding Convolutional-Coded Data.................................................................................. 7-22 

7.6.3 BER Calculation for Convolutional Coding ........................................................................ 7-22 

7.6.4 BER Curves ......................................................................................................................... 7-25 

7.7 CPM............................................................................................................................................ 7-25 

7.7.1 Description........................................................................................................................... 7-25 

v 
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7.7.2 BER Curves ......................................................................................................................... 7-25 

7.8 CONCATENATED CODES......................................................................................................7-29 

SECTION 8 – SOURCE DECODER...................................................................................................... 8-1

8.1 MULTI-BIT SAMPLING.............................................................................................................8-1 

8.1.1 Description............................................................................................................................. 8-1 

8.1.2 Output Signal-to-Noise ..........................................................................................................8-3 

8.2 SINGLE-BIT SAMPLING........................................................................................................... 8-5 

8.2.1 Description............................................................................................................................. 8-5 

8.2.2 Output Signal-to-Noise ..........................................................................................................8-6 

SECTION 9 – SAMPLE RECEIVER ANALYSIS PROBLEMS .......................................................... 9-1 

9.1 RECEIVER PERFORMANCE ANALYSIS EXAMPLE 1.........................................................9-1 

9.1.1 Approach................................................................................................................................ 9-1 

9.1.2 Models.................................................................................................................................... 9-1 

9.1.3 Calculate Input Signal Requirement for System 1................................................................. 9-2 

9.1.4 Calculate Input Signal Requirement for System 2................................................................. 9-2 

9.1.5 Comparing the Two Implementations ...................................................................................9-2 

9.2 RECEIVER PERFORMANCE ANALYSIS EXAMPLE 2.........................................................9-3 

9.2.1 Approach................................................................................................................................ 9-3 

9.2.2 Operating Environment..........................................................................................................9-4 

9.2.3 Equipment Characteristics (for the Desired Signal) .............................................................. 9-4 

9.2.4 Required Performance ...........................................................................................................9-4 

9.2.5 Received Signal Power Calculation.......................................................................................9-5 

9.2.6 Comparison of Received-Signal Power With Receiver Sensitivity.......................................9-5 

9.2.7 Noise Calculation................................................................................................................... 9-5 

9.2.8 Interference Calculation.........................................................................................................9-6 

9.2.9 Remediation ........................................................................................................................... 9-7 

Figures 

2-1. Basic Link Budget Model................................................................................................................2-4 

2-2. More Detailed Link Budget Model .................................................................................................2-4 

2-3. System Performance Model.............................................................................................................2-7 

2-4. Receiver Performance Model ........................................................................................................2-14 

5-1. AI vs. (S/N)in Curves for AM Voice Receiver with ASK Interference...........................................5-7 

5-2. AI vs. (S/N)in Curves for AM Voice Receiver with FSK Interference ...........................................5-8 

5-3. AI vs. (S/N)in Curves for SSB Voice Receiver with ASK Interference ........................................5-11 

5-4. AI vs. (S/N)in Curves for SSB Voice Receiver with FSK Interference.........................................5-12 

5-5. AI vs. (S/N)in Curves for FM Voice Receiver with ASK Interference .........................................5-17 

5-6. AI vs. (S/N)in Curves for FM Voice Receiver with FSK Interference ..........................................5-18 

vi 
 JSC-CR-06-072 

5-7. TASO Score vs. (S/I)in Curves for Broadcast TV Receiver..........................................................5-20 

6-1. Analytical and Simulation BER Curves for PSK Receiver with M = 8 ..........................................6-7 

6-2. BER vs. Eb/No Curves for PSK Receiver (M = 2) with CW Interference .....................................6-11 

6-3. BER vs. Eb/No Curves for PSK Receiver (M = 4) with CW Interference .....................................6-12 

6-4. BER vs. Eb/No Curves for PSK Receiver (M = 8) with CW Interference .....................................6-13 

6-5. BER vs. Eb/No Curves for PSK Receiver (M = 16) with CW Interference ...................................6-14 

6-6. BER vs. Eb/No Curves for DPSK Receiver (M = 2) with CW Interference ..................................6-17 

6-7. BER vs. Eb/No Curves for DPSK Receiver (M = 4) with CW Interference ..................................6-18 

6-8. BER vs. Eb/No Curves for DPSK Receiver (M = 8) with CW Interference ..................................6-19 

6-9. BER vs. Eb/No Curves for Coherent FSK Receiver (M = 2) with CW Interference......................6-21 

6-10. BER vs. Eb/No Curves for Noncoherent FSK Receiver (M = 2) with CW Interference..............6-22 

6-11. BER vs. Eb/No Curves for Noncoherent FSK Receiver (M = 4) with CW Interference..............6-23 

6-12. BER vs. Eb/No Curves for Noncoherent FSK Receiver (M = 8) with CW Interference..............6-24 

6-13. BER Curves for an MSK Receiver with CW Interference..........................................................6-27 

6-14. BER vs. Eb/No Curves for QBL-MSK Receiver with CW Interference......................................6-30 

6-15. BER vs. Eb/No Curves for Coherent Unipolar ASK Receiver (M = 2) with CW Interference....6-32 

6-16. BER vs. Eb/No Curves for Coherent Unipolar ASK Receiver (M = 4) with CW Interference....6-33 

6-17. BER vs. Eb/No Curves for Coherent Bipolar ASK Receiver (M = 4) with CW Interference......6-34 

6-18. BER vs. Eb/No Curves for Coherent Bipolar ASK Receiver (M = 8) with CW Interference......6-35 

6-19. BER vs. Eb/No Curves for Noncoherent ASK Receiver..............................................................6-36 

6-20. BER vs. Eb/No Curves for QAM Receiver (M = 16) with CW Interference ...............................6-39 

6-21. BER vs. Eb/No Curves for QAM Receiver (M = 64) with CW Interference ...............................6-40 

7-1. Output BER vs. Input BER Curve for Hard-decision (255, 239) BCH Decoder............................7-8 

7-2. Output BER vs. Input BER Curve for Hard-decision (8, 4) Hamming Decoder ..........................7-10 

7-3. BER vs. Eb/No Curves for Soft-decision (8, 4) Hamming Decoder ..............................................7-11 

7-4. Output BER vs. Input BER Curve for Hard-decision (64, 6) Hadamard Decoder........................7-13 

7-5. BER vs. Eb/No Curves for Soft-decision (64, 6) Hadamard Decoder............................................7-14 

7-6. Output BER vs. Input BER Curve for Hard-decision (24, 12) Golay Decoder ............................7-15 

7-7. BER vs. Eb/No Curves for Soft-decision (24, 12) Golay Decoder.................................................7-16 

7-8. Output BER vs. Input BER Curve for Hard-decision (255, 223) RS Decoder .............................7-18 

7-9. Pulsed Interference Without Interleaving......................................................................................7-19 

7-10. Pulsed Interference With Interleaving.........................................................................................7-19 

7-11. A (1/2, 3) Convolutional Encoder ...............................................................................................7-21 

7-12. BER vs. Eb/No Curves for Soft-decision (1/2, 7) Viterbi Decoder..............................................7-26 

7-13. BER vs. Eb/No Curves for Binary CPM Decoder (Modulation Index 1/2)..................................7-27 

7-14. BER vs. Eb/No Curves for 4-ary CPM Decoder (Modulation Index 1/6)....................................7-28 

7-15. Concatenated Codes ....................................................................................................................7-29 

8-1. Block Diagram for PCM ADC and DAC........................................................................................8-2 

8-2. Output S/N vs. Input BER for PCM DAC.......................................................................................8-4 

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8-3. Block Diagram for DPCM Encoder ................................................................................................8-6 

8-4. Output S/N vs. Input BER for 16-kbit CVSD .................................................................................8-7

Tables 

3-1. RF/IF Output Waveform for Various Interfering Input Signals......................................................3-7 

5-1. TASO Score Definition .................................................................................................................5-19 

7-1. Linear, Systematic (7, 4) Block Code..............................................................................................7-4 

7-2. Performance of (7, 4) Linear Block Code With t = 1......................................................................7-6 

7-3. BCH Code Parameters for n ≤ 63....................................................................................................7-7 

7-4. Rate R = 1/2 Convolutional Codes................................................................................................7-21 

7-5. Transfer Functions for R = 1/2 Convolutional Codes...................................................................7-23

7-6. Derivative Polynomials for R= 1/2 Convolutional Codes.............................................................7-23 

viii 
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GLOSSARY 

ADC Analog-to-Digital Converter

ADM Adaptive Delta Modulation 

AI Articulation Index

AM Amplitude Modulation

AS Articulation Score

ASK Amplitude-Shift Keying

BCH Bose-Chaudhurie-Hocquenghem

BER Bit Error Rate 

BPSK Binary Phase-Shift Keying 

CFSK Coherent Frequency-Shift Keying 

CPM Continuous-Phase Modulation

CVSD Continuously Variable Slope Delta 

CW Continuous Wave

DAC Digital-to-Analog Converter

DM Delta Modulation

DPCM Differential Pulse Code Modulation 

DPSK Differential Phase-Shift Keying 

DSB Double Sideband

Eb/No Ratio of bit energy to noise power density 

EIRP Effective Isotropic Radiated Power 

EMC Electromagnetic Compatibility

FDR Frequency-Dependent Rejection

FEC Forward Error Correction 

FM Frequency Modulation

FSK Frequency-Shift Keying

IF Intermediate Frequency

MSK Minimum-Shift Keying

NCFSK Non-coherent Frequency-Shift Keying 

NTIA National Telecommunications and Information Administration 

OQPSK Offset Quadrature Phase-Shift Keying 

PCM Pulse Code Modulation 

PN Pseudo-noise

 ix 
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PRI Pulse Repetition Interval 

PSD Power Spectral Density 

PSK Phase-Shift Keying

QAM Quadrature Amplitude-Modulation

QBL-MSK Quasi-Bandlimited Minimum-Shift Keying

QPSK Quadrature Phase-Shift Keying 

RF Radio Frequency 

RMS Root-mean-square

RS Reed-Solomon

SER Symbol Error Rate 

S/I Signal-to-Interference Ratio

S/N Signal-to-Noise Ratio

SSB Single Sideband

TASO Television Allocation Study Organization 

WER Word Error Rate

x 
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SECTION 1 – INTRODUCTION 

1.1 BACKGROUND 

In May 2003, President Bush established the Spectrum Policy Initiative to promote the development and 

implementation of a United States spectrum policy for the 21st century. In response to the Spectrum

Policy Initiative, the Secretary of Commerce established a Federal Government Spectrum Task Force 

and initiated a series of public meetings to address policies affecting spectrum use by the federal 

government, state, and local governments, and the private sector. The recommendations resulting from

these activities were included in two reports released by the Secretary of Commerce in June 2004.1-1 In 

November 2004, the President directed the federal agencies to develop a plan to implement the 24 

recommendations contained in the reports.1-2 One of the recommendations directed the National 

Telecommunications and Information Administration (NTIA) to develop a handbook documenting best 

practices in spectrum engineering that will be recognized by all regulatory authorities in the United 

States. The Best Practices Handbook will address the electromagnetic compatibility (EMC) analysis 

process. Establishing receiver performance degradation levels is required to perform an EMC analysis. 

The NTIA requested that the Joint Spectrum Center develop a communications receiver performance 

degradation handbook that could be included as part of the Best Practices Handbook. 

1.2 PURPOSE OF THE HANDBOOK 

The purpose of the handbook is to provide the radio frequency (RF) analyst with the capability to 

calculate the effects of noise and interference on RF communications receivers. A receiver is modeled 

as a sequence of modules. Each module has a transfer function that relates the module outputs to the 

module inputs. By consecutively analyzing each module in the sequence, the analyst can then relate the 

receiver outputs (performance) to the receiver inputs (signal characteristics). 

The first version of this document includes only transfer functions that were immediately available at the 

time of development. Future versions of the handbook could include an expanded set of functions. 

 

1-1 Spectrum Policy for the 21st Century – The President’s Spectrum Policy Initiative: Report 1, Department of

Commerce (June 2004), at http://www.ntia.doc.gov/reports/specpolini/presspecpolini_report1_06242004.htm; 

Spectrum Policy for the 21st Century – The President’s Spectrum Policy Initiative: Report 2, Department of

Commerce (June 2004), at http://www.ntia.doc.gov/reports.html. 1-2 White House Executive Memorandum, Subject: Improving Spectrum Management for the 21st Century (23 

November 2004). The latest released document for this subject is Spectrum Management for the 21st Century –

Plan to Implement Recommendations of the President’s Spectrum Policy Initiative, Department of Commerce 

(March 2006), at http://www.ntia.doc.gov/reports.html. 

1-1 
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1.3 OVERVIEW OF HANDBOOK CONTENTS 

Section 2 describes the general procedures for performing a communications receiver performance 

analysis. It introduces the fundamental concepts and describes the receiver model. It does not include 

detailed descriptions for the individual modules within the receiver. 

The first module in the receiver is designed to amplify the desired signal, convert it to an intermediate 

frequency (IF), and filter out some interference and noise. Section 3 provides a detailed description of

the RF/IF section. It also specifies how interfering signals may be changed by the filters. 

Section 4 describes the despreader module in spread spectrum receivers. It specifies how the spread 

spectrum processing gain can be calculated. It also provides information that can be used to analyze 

spread spectrum multiple-access systems. 

Sections 5 and 6 describe the demodulator module. They provide plots that characterize performance as 

a function of the input power ratios. Analog voice and broadcast television receivers are considered in 

Section 5 and digital receivers are considered in Section 6. 

Section 7 describes forward error correction (FEC) decoders, which use redundancy bits to reduce the 

bit-error rate (BER) by correcting some of the bit errors introduced by interference and noise. It 

provides plots of output BER vs. input BER for several types of FEC decoders. 

Section 8 describes source decoders, which convert the information bit sequence to the final format at 

the receiver output. For a digital voice system, this format is an analog voice waveform. Section 8 

provides plots that relate output signal quality to the input BER. 

Section 9 presents two examples that demonstrate how this handbook can be used to perform a receiver 

analysis. 

1.4 HOW TO USE THE HANDBOOK 

The first step in using this handbook is to learn about the communications receiver performance analysis 

model discussed in Section 2. That section introduces some basic concepts and terminology and sets the 

context in which a receiver analysis may be performed. It also discusses the limitations and scope of the 

model. It displays the general structure of the receiver model as a sequence of distinct modules, each of 

which performs a specific signal-processing function. 

The second step in using the handbook is to actually perform a receiver analysis. The analysis objective 

is either to determine a receiver performance measure value (such as BER) for a given set of input signal 

1-2 
JSC-CR-06-072 

conditions or, conversely, to determine the input signal conditions that would yield a given performance 

measure value. 

The receiver modules are represented in the handbook by transfer functions, which are organized by 

module in Sections 3 through 8. Section 2.4.5 lists the specific steps to follow in using these transfer 

functions to perform an analysis. Section 9 gives two receiver analysis examples. 

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SECTION 2 – ANALYSIS PROCEDURES 

This section describes the procedures for performing a communications receiver performance analysis. 

To set the context, it specifies types of RF systems and types of RF signals. Then it introduces some

basic analysis concepts with a simple link budget analysis model. Finally, it describes the detailed 

model that is the subject of this handbook: the receiver performance analysis model. 

2.1 SYSTEM TYPES 

The focus of this handbook is on RF communications systems. The RF spectrum can be used for many 

purposes other than what is usually thought of as communications. Although many of the analysis 

results presented in this handbook may be applied to RF systems in general, all discussions and data will 

be presented in a communications context. 

2.1.1 Communications 

RF communications systems are designed to send a message from one point to another by converting the 

message to an RF signal and transmitting that signal. 

2.1.1.1 Point-to-point

The most basic RF communications system is composed of a transmitter at a given location, a receiver at 

a given location, and the channel through which the RF signal propagates. Such a system is called a 

point-to-point system because the signal travels between two fixed points. This permits directional 

antennas to be used for both the transmitter and receiver. These antennas can reduce the output power 

required from the transmitter by several orders of magnitude. A point-to-point system is also called a 

communications link. 

2.1.1.2 Point-to-multipoint 

Another common type of RF communications system has one transmitter and many receivers, which are 

located at different points within the transmission range of the transmitter. In this case, the transmitter 

antenna is often non-directional, and must therefore transmit enough power to cover the appropriate 

area. Such a system is called a point-to-multipoint system. Each transmitter-receiver pair constitutes an 

independent communications link, and a link analysis may be performed for each one. 

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2.1.1.3 Mobile 

Mobile systems require that the antennas not be constrained to favor one fixed direction over another. 

This can be accomplished very simply by utilizing omnidirectional antennas. Alternatively, tracking 

antennas can be used to adapt to changes in the direction of propagation. This adaptation can be 

accomplished physically, by moving the antennas, or electrically, by means of steerable-beam arrays. 

2.1.2 Non-communications 

There are many RF non-communications systems that are not designed to send a message from one 

point to another. RF technology may be used to determine geographic location. Examples include 

RADAR, global positioning system, and navigation systems. A special type of RADAR can be used for 

spatial data collection, particularly terrain data. Radio frequency identification tags are becoming 

commonplace, as a means of tracking the location of various objects. Remote control can be 

implemented with RF technology, enabling operators to perform tasks at inaccessible or inconvenient 

locations. 

2.2 SIGNAL TYPES 

RF signals may be classified in several ways. From the perspective of a receiver, an incoming signal 

may be either desired or undesired. An RF signal may also be classified according to its waveform 

properties. These properties are responsible for the ways in which the signal interacts with the receiver 

components. 

2.2.1 Noise 

Noise is present in every practical RF system component, and in every propagation channel. Noise 

imposes practical limits on RF communications, because it is always necessary for the receiver to

distinguish the desired signal from the noise. Consequently, RF analysis results are often expressed as 

the ratio of the desired signal to the noise. Noise is modeled in the time domain as a signal whose 

amplitude varies according to the Gaussian probability distribution. Noise is modeled in the frequency 

domain as a signal that has a constant magnitude at all frequencies within the receiver band. 

2.2.2 Continuous-Wave 

A continuous-wave (CW) signal is a sinusoid. Its properties are, in a sense, the opposite of the 

properties of a noise signal. A CW signal has a constant-amplitude envelope, and only one frequency. 

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CW signals are used as RF carriers, beacons, and reference signals. Interfering (undesired) signals are 

sometimes modeled as CW signals. 

2.2.3 Modulated 

One signal can be used to control the amplitude, frequency, phase, or other property of another signal 

through a process called modulation. Communications signals are typically modulated signals, since the 

modulation process can be used to convey information. There are many specific modulation schemes. 

Each has certain advantages and disadvantages with respect to system performance. This handbook 

includes analysis results for many common modulation schemes. Modulated signals occupy more RF 

spectrum than CW signals; the emission bandwidth depends on the particular modulation technique (and 

on other characteristics such as data rate). 

2.2.4 Intermittent 

A signal may periodically or randomly exhibit large amplitude fluctuations for various reasons. This 

intermittent behavior may, in fact, constitute the most important property of a signal. Pulsed and 

frequency-hopping systems are examples of intermittent signal sources. For intermittent signals, it is not 

generally sufficient to consider properties (such as power) that have been averaged over a long period of 

time. Instead, it is necessary to consider the temporal fluctuation of signal properties and to include the 

effects of those fluctuations in the analysis. 

2.3 LINK BUDGET ANALYSIS 

A basic link budget analysis technique assumes that a communications link can be modeled as a series 

of independent modules originating at the transmitter and terminating at the receiver input. Each 

module is represented by its output power, gain, or loss. Link budget analysis determines the effect of 

each module on the desired or undesired signal power at the receiver input, without tracing a signal 

through the various components within the receiver. If the analysis shows the signal power to be too 

low for a given link, then more power must be budgeted or allocated for that receiver (e.g., by moving 

the antennas closer together). 

2.3.1 Link Budget Model 

The most basic model is composed of a transmitter and receiver linked by a propagation channel, as 

shown in Figure 2-1. The transmitter signal is represented by the transmitted power, which is delivered 

to the propagation channel. The channel subjects the signal to propagation loss, which is a reduction in 

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power of the transmitted signal. The received signal is simply the transmitted signal, reduced in 

magnitude by the propagation loss. The model in Figure 2-1 is surprisingly useful, given its simplicity. 

If there are multiple transmitters (a desired transmitter and one or more interfering transmitters), then the 

model in Figure 2-1 is used for each transmitter. For the desired transmitter, it yields the desired signal 

power at the receiver input. For an interfering transmitter, it yields the interfering signal power at the 

receiver input. 

Transmitter Propagation

Channel Receiver

Transmitted

Power

Received

Power Propagation

Loss

Figure 2-1. Basic Link Budget Model 

The model in Figure 2-1 makes no mention of antennas, which are assumed to be lumped together with 

other modules. This may not be acceptable in some cases because it may be important to consider the 

antenna effects separately. Figure 2-2 shows a more detailed model. The transmitter and receiver 

antennas are represented by separate modules, permitting their contribution to the overall performance to 

be considered separately. 

Transmitter Transmitter

Antenna

Propagation

Channel

Receiver

Antenna Receiver

Transmitter

Power

Received

An Power tenna

Gain

Propagation

Loss

Antenna

Gain

Figure 2-2. More Detailed Link Budget Model 

Transmitter. The only transmitter characteristic considered is the transmitter output power PT, which is 

typically expressed in decibels referenced to power units, such as decibels relative to one milliwatt 

(dBm) or decibels relative to one watt (dBW). 

Transmitter Antenna. The effect of the transmitter antenna is expressed as a power gain GT with respect 

to an isotropic antenna, expressed in units of dBi. Antennas radiate in all directions, but many antennas 

are designed to favor one or more particular directions. For these directional antennas, the gain will 

depend on the orientation of the antenna with respect to the propagation path. An antenna pattern, 

which shows the gain in all directions, may be used to determine the gain for a particular link. Antenna 

patterns usually describe the far-field radiation pattern. 

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Effective Isotropic Radiated Power (EIRP). This quantity, which combines the transmitter power with 

the transmitter antenna gain, is often used to simplify the analysis. If the antenna is directional, the 

EIRP will be valid only in the direction corresponding to the antenna gain that was used in determining 

the EIRP. The unit of EIRP is the same as for the transmitter power: dBm, dBW, etc. 

Propagation Loss. The propagation loss is expressed as the ratio of the transmitted power to the received 

power, not including the antenna gains. Since this ratio is always greater than 1, the propagation loss in 

dB is always a positive quantity. 

Receiver Antenna. The gain GR of the receiver antenna with respect to an isotropic antenna is expressed 

in dBi. The same considerations of directionality that applied to the transmitter antenna also apply to the 

receiver antenna. 

2.3.2 Link Budget Equation 

A typical application of the link budget concept is the calculation of the received-signal power for a lineof-sight communications link: 

= + + − 20 log ( ) − 20 log ( ) + 27.6

= + + −

P G G d f

P P G G L

T T R

R T T R

(2-1) 

where 

PR = signal power at receiver input, in dBm 

PT = signal power at transmitter output, in dBm

GT = transmitter antenna gain, in dBi 

GR = receiver antenna gain, in dBi 

L = free-space propagation loss, in dB 

d = distance between antennas, in meters

f = transmitter frequency, in MHz 

Equation 2-1 assumes free-space propagation. If the EIRP is specified instead of the transmitter power, 

then the following relationship is used: 

PR = EIRP + GR − L

(2-2) 

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2.3.3 Link Budget Performance Measures 

The received-signal power (of the desired signal) may be used to predict the overall quality of the 

communications link. To make such a prediction, it is necessary to have a specification of required 

performance to which the calculated quantity can be compared. These specifications depend on many 

factors, will vary from system to system, and should include external noise and interference. 

A very basic specification is the minimum signal that the receiver can detect. This may be called the 

minimum discernable signal or the minimum detectable signal. However, the fact that a signal is strong 

enough to be discerned or detected does not imply that it is strong enough for the message to be 

accurately recovered. 

A more useful specification is the receiver sensitivity. This is the minimum received signal level that 

the receiver can use to produce a certain required output. The required output will usually be a 

particular value of BER or signal-to-noise ratio (S/N). 

2.3.4 Limitations of Link Budget Analysis 

The simple link-budget analysis technique presented here is often used because of its simplicity. 

However, that simplicity is based on several assumptions which limit the usefulness of the technique. 

It is assumed that average power provides a realistic and meaningful characterization of the input signal. 

This is valid if each of the independent variables in the analysis (e.g., the independent variables in 

Equation 2-1) is constant in time. Many systems, however, do exhibit time variation of one or more of 

these variables. 

Equation 2-1 assumes free-space propagation. The effects of the propagation environment, except for 

distance, are ignored. This equation can be modified to accommodate more complex types of 

propagation loss. The propagation loss calculation for many practical systems requires the availability 

of sophisticated propagation models. 

A link-budget analysis does not directly address the performance of the receiver. Even when the 

receiver sensitivity is used as a performance measure, the corresponding output BER or S/N may be 

unknown or may be inappropriate for the given context. For example, in the presence of an interfering 

signal the desired signal level must generally exceed the receiver sensitivity. The processing that takes 

place once the signals have entered the receiver determines the fidelity and usefulness of the final 

output. It is this final set of signal-processing operations within the receiver that this handbook 

addresses in detail. 

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2.4 RECEIVER PERFORMANCE ANALYSIS 

In contrast to the link-budget analysis, a system performance analysis directly addresses the performance 

of the receiver. The system performance model is divided into three parts: the transmitter model, the 

RF channel model, and the receiver performance model. This handbook presents the methodology for 

performing an analysis with the receiver performance model. 

2.4.1 System Performance Model 

The system performance model is shown in Figure 2-3. The transmitter and receiver are modeled as 

systems which in turn are composed of a number of modules. These modules – or functional blocks – 

represent distinct signal-processing operations. The transmitter modules systematically prepare the 

information signal for propagation through the RF channel. The receiver essentially reverses the 

operations performed by the transmitter to obtain the information. The antennas have been included in 

the RF channel which, in addition, contains an interference source. The interference propagates through 

its own RF channel. The interference RF channel is not shown in the illustration. 

Source

Encoder

Error Correction

Coder Modulator Spreader RF/IFSection

Transmitter

Output

Power

Bandwidth

Modulation Type

Transmitter

text

Propagation

Channel

Transmitter

Antenna

Receiver

Antenna

Interference

RF Channel

Source

Decoder

Error Correction

Decoder Despreader Demodulator RF/IFSection

Receiver

Input

Desired Signal Power

Undesired Signal Power

Receiver

Figure 2-3. System Performance Model 

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One benefit of placing the antenna models in the RF channel model is that the effects of site location are 

grouped together. These effects include antenna orientation, separation distances, and environmental 

phenomena such as atmospheric absorption and ground conductivity. This organization permits analysts 

who specialize in RF propagation to focus on that part of the communication link. The communications 

system model in Figure 2-3 has proven to be a very practical and useful representation. 

2.4.1.1 Transmitter 

As modeled, the transmitter has five modules. Not all transmitters will have all five modules. For 

example, the transmitter of a purely analog system has only two modules: the RF/IF section and the 

modulator. 

Source Encoder. The source encoder converts the source into a sequence of bits at a certain data rate. 

If the source is analog, an analog-to-digital converter (ADC) samples and encodes the analog waveform. 

If the source is digital, the source encoder reformats and retimes the input data if necessary. The output 

of the source encoder is a sequence of information bits, at the information data rate. 

FEC Encoder. The FEC encoder encodes the data from the source encoder to meet requirements 

dictated by the communication channel. The FEC encoder adds code bits to the information bit 

sequence. In many cases, this requires that the data rate be increased to accommodate the extra bits. 

The output of the FEC encoder is a sequence of code bits, at the coded data rate. 

Modulator. The modulator creates an analog waveform (carrier) whose properties vary in accordance 

with the input waveform or bit sequence. The amplitude, frequency, and phase of the carrier may be 

modified by the modulation process. There are a great many modulation schemes. The output of the 

modulator is a modulated carrier waveform. The frequency of this waveform is usually not suitable for 

RF transmission, and will have to be changed (usually increased) in subsequent processing. 

Spreader. The digital modulator output waveform will have a bandwidth that is a function of the coded 

data rate and the modulation type. The spreader creates a so-called “spread-spectrum” signal, which 

increases this bandwidth. There are two common approaches to spectrum spreading. Frequencyhopping utilizes a number of different carrier frequencies rather than just one. These carriers span a 

frequency range that is much larger than if the single carrier were used. Direct sequence converts the 

input data rate to a much higher output data rate. The resulting waveform has a much larger bandwidth 

than if the lower data rate were used.

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RF/IF Section. The RF/IF section provides frequency translation, filtering, and amplification to prepare 

the signal for transmission. The output of this section is delivered to the transmitter antenna by a cable, 

waveguide, or some other conductor. 

2.4.1.2 RF Channel 

Transmitter Antenna. The transmitter antenna may be designed to focus RF energy in a particular 

direction for a point-to-point system, or it may be omnidirectional, to support point-to-multipoint and 

mobile communications. The transmitter antenna adds gain (or loss) to the transmitter output. 

Propagation Channel. The most basic propagation channel is free space – the transmitted signal is 

attenuated only by distance. A more realistic propagation channel could include the effects of natural 

and man-made obstructions, atmospheric effects, ground effects, and other effects. Propagation 

modeling can be very complex. Many models have been developed to simulate the propagation of RF 

energy in various environments. The use of these models is somewhat specialized, and is not covered in 

this handbook. The ultimate purpose of a propagation model is to predict the magnitude (and possibly 

other characteristics such as phase) of the transmitted signal at the receiver location.

Interference. The interference source is typically another transmitter in the environment. To simplify 

Figure 2-3, it is shown as a single block that feeds the receiver antenna block. The actual transmitter, of

course, creates a signal in its own modules and sends that signal through its antenna and through a 

separate propagation channel. In addition, there may be multiple interfering transmitters. 

Receiver Antenna. The purpose of the receiver antenna is to collect sufficient energy from the 

transmitted signal so that the transmitted information can be extracted by the receiver system. 

Depending on its design and orientation, the receiver antenna may add gain or loss to the signal. 

2.4.1.3 Receiver 

The receiver takes its input from the receiver antenna, and performs processing complementary to that 

done by the transmitter. 

RF/IF Section. The RF/IF section converts the frequency of the received signal to an IF that is more 

suitable for signal processing. It also filters the composite signal to reduce the undesired signal power. 

Despreader. The despreader, in the case of a frequency-hopping system, resolves the multiple-carrier

input so that a single IF is obtained. In the case of a direct-sequence system, the despreader removes the 

spreading code from the desired signal. 

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Demodulator. A digital demodulator converts the input waveform to a bit sequence. If error correction 

coding has been applied, this is the coded bit sequence, at the coded data rate. An analog demodulator 

produces a replica of the original baseband analog waveform (but with some noise, interference, and 

distortion). 

FEC Decoder. The FEC decoder performs the digital processing necessary to reconstruct the original 

information bit sequence. The output is the information bit sequence, at the information bit rate. This 

bit sequence may contain residual errors. 

Source Decoder. The source decoder converts the information bit sequence to the format required by 

the user of the receiver. If this format is analog, a digital-to-analog converter (DAC) will be employed 

at this stage. The output of the source decoder is the output of the receiver. In general, it will differ 

somewhat from what was transmitted due to noise and interference. This difference can be quantified, 

and is a measure of the overall system performance. 

2.4.2 System Performance Measures 

The goal of an RF analysis is to predict the performance of a system quantitatively. The performance 

prediction can then be compared to certain performance goals to obtain an assessment of the system 

operation. There are several useful measures of performance. 

2.4.2.1 Signal-to-Noise Power Ratio 

The S/N is calculated by determining the desired-signal power S and the noise power N at some point in 

the system. The S/N is always expressed in dB. Note that if S and N are in decibel units, such as dBm,

the signal-to-noise "ratio" is actually S - N. Depending on the system and the application, target 

performance goals for S/N might range from 10 to 30 dB. If noise-like interference is present, then the 

variable N represents the sum of the interference power and the noise power (added in non-logarithmic 

units such as mW). 

2.4.2.2 Signal-to-Interference Power Ratio 

The signal-to-interference power ratio (S/I) is calculated when there is an undesired signal which cannot 

be adequately modeled as additional noise. For example, in addition to noise there may also be a CW

interfering signal. 

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2.4.2.3 Bit Error Rate 

The BER, which obviously applies to digital systems, can be interpreted two ways. It is the probability 

of a particular bit being wrong, and it is also the ratio of bit errors to total bits received in the long term. 

The latter interpretation is useful when the error-causing process is constant over the analysis interval. 

The target BER after error correction is often between 10-3 and 10-6. 

2.4.2.4 Subjective Performance Measures 

Measures of signal power or BER do not directly address the quality of communication as perceived by 

users of the system. Some subjective performance measures have been defined that do address that 

quality of communication. For voice communications, the articulation score (AS) is defined as the 

percentage of words, phrases, sentences, or other message elements correctly identified by a listener 

panel. The Television Allocation Study Organization (TASO) score is a similar evaluation of television 

picture quality by a viewer panel. The articulation index (AI) is a calculated quantity that is designed to 

be a predictor of voice intelligibility. 

It is possible to establish a correspondence between objective performance and subjective performance 

by extensive, careful testing and polling. It is very difficult, however, to extrapolate subjective test 

results beyond the parameter set used for the tests. 

2.4.2.5 Time-varying Performance 

When reporting results in terms of S/N or BER, it may be assumed that these quantities are relatively 

constant. This implies, of course, that the phenomena responsible for these measures of performance are 

relatively constant. If this is not the case, then the temporal variations should be taken into account. 

These variations are often expressed statistically. For example, a quality-of-service target might be a 

BER of less than 10-5 at least 90% of the time, and a BER of greater than 10-3 no more than 1% of the 

time. 

2.4.3 System Performance Analysis Limitations 

In Section 2.3.4, the limitations of a link budget analysis (exclusion of systems with temporal variations, 

unsophisticated propagation model, no detailed receiver model) were presented. The first two of those 

limitations are still concerns in a system performance analysis. A system performance analysis does 

have detailed receiver modeling, but some effects occur within a receiver that are not included; namely, 

the nonlinear effects that occur when an interfering signal is very strong. 

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2.4.3.1 Temporal Variations 

The analysis techniques and transfer functions presented in the handbook are useful for systems whose 

performance does not vary significantly with time. These are systems to which non-time-varying 

quantities such as average power, bit-error rate, S/N, and S/I may be assigned. If a system does exhibit 

significant temporal variation, then the nature of the variation must be considered. Performance 

measures must be revised accordingly. Two types of temporal variations are fading and intermittent

interference.

Fading. The term fading applies to unexpectedly large variations in the desired signal power at the 

receiver. The cause of the variation may be understood, but may be impractical to model. For example, 

fading may be caused by multipath interference in mobile communications. Small changes in position 

can result in large changes in the way the signal replicas combine at the receiver, leading to occasional 

periods of weak reception known as fades. 

Fading may be accounted for very simply by specifying a fade margin, which is an increase in the

magnitude of received-signal power that is required for proper system operation. The size of the fade 

margin may be determined experimentally by field measurement, or may be based on experiences with 

similar systems in similar environments. 

Outages. An outage is a period of time during which system operation is so compromised that for all 

practical purposes it does not work at all. It is therefore not helpful to attempt to quantify receiver 

performance during an outage. Instead, the process responsible for the outage should be analyzed, and 

the effects on the system modeled statistically. An outage may be a single, catastrophic event, such as a 

lightning strike. In this case, a typical performance measure is simply the probability of occurrence. 

Intermittent interference tends to produce repetitive outages. For example, each pulse from a pulsed 

radar or each co-channel hop from a frequency hopping transmitter may produce an outage. For 

repetitive outages, the performance measure must be tailored to the system that is affected. Such 

performance measures typically specify the required performance during the “quiescent” periods 

between outages, as well as the maximum frequency or duration of the outages. For example, a system

may be required to operate below a certain BER a certain percentage of the time. The rest of the time, 

the system may experience outages during which communications will be interrupted. 

Intermittent interference may also result in periods of partially compromised performance, rather than 

full outages. A statistical analysis is generally required to evaluate such systems. 

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2.4.3.2 Complex RF Propagation Environments 

In any analysis one can assume a free-space propagation path loss. In many cases, this is not a valid 

assumption. The propagation environment includes natural and manmade obstructions such as terrain, 

foliage, and buildings. These obstructions affect propagation by blocking, diffracting, absorbing, 

scattering, and reflecting RF energy. The atmosphere also affects propagation through refraction, 

absorption, ducting, and scattering. These environmental effects are frequency dependent and, in some

cases, intermittent in nature. Nevertheless, loss from propagation is usually the largest factor in the link 

budget – by many orders of magnitude. 

There are many analytical and simulation-based RF propagation models available. These should be 

employed when needed to predict the RF propagation loss and, in turn, the received-signal power. 

2.4.3.3 Collocated RF Equipment 

A cosite environment is one in which interference power levels are so high that nonlinear effects such as 

desensitization and intermodulation must be considered. The interfering signal power level may be high 

because the transmitter and receiver antennas are close together (collocated). A typical cosite 

environment is a platform such as a ship, airplane, or tower on which multiple antennas are located. 

As an example, consider the case in which a strong interfering signal is far off-tune from the receiver 

frequency. Because it is far off-tune, it is attenuated so much by the receiver filters that the residual 

signal has no effect on the receiver. However, before it enters those filters it passes through the RF

amplifier. If its power is outside of the intended (linear) operating range of the amplifier, it will 

generally desensitize the amplifier; that is, reduce the amplifier gain. This, in turn, reduces 

amplification of the desired signal power so that the desired signal gets lost in the noise. Thus, the 

positive effects of filtering have been defeated by the nonlinear effect. 

Cosite effects are not included in this handbook. For cases in which filtering plays a significant role in 

determining an interference threshold, the analyst should ensure that the signal power is sufficiently low 

that nonlinear effects will not occur. If nonlinear effects are possible, cosite analysis techniques should 

be applied to adjust the frequency and/or power level of the interfering signal that is input to the RF/IF 

section (see Figure 2-3). 

2.4.4 Receiver Performance Analysis Types 

There are generally three techniques – analytical, measurement, and simulation – that can be used to 

assess the performance of a receiver.

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In the analytical technique, the operations of receiver modules are represented by equations that can be 

solved in closed form or by numerical methods. Many of the transfer functions in the handbook were 

obtained analytically. 

In the measurement technique, receiver equipment is acquired and subjected to laboratory measurement. 

This technique is useful for investigating the performance of specific hardware, but it is lacking in 

generality. Measurement was not used directly in developing the receiver transfer functions, but many 

of the results have been confirmed by measurement. 

In the simulation technique, software generates samples of the signal waveforms and processes these 

samples through the simulated receiver components. Simulated measurement devices then determine 

performance measures such as BER. Software simulation of RF systems is useful for systems that are 

too difficult to characterize analytically. Simulation permits a fine degree of control over system

parameters. Many of the results in the handbook were obtained by simulation. 

These techniques can be combined. For example, measurement data for a functional block can be 

entered into a simulation. A hardware module can even be used directly as a functional block in a 

simulation system. This technique may be used to test specific modules in the controlled, repeatable 

environment that a simulation system provides. 

2.4.5 Receiver Performance Analysis Steps 

A receiver performance analysis is that part of the system performance analysis that focuses on 

processing within the receiver. The communications receiver model is shown in Figure 2-4, which has 

been extracted from Figure 2-3. Each processing block is represented by a transfer function that 

expresses the output of the block in terms of the input. 

Source

Decoder

Error Correction

Decoder Despreader Demodulator RF/IFSection

Receiver

Input

Desired Signal Power

Undesired Signal Power

Receiver

Figure 2-4. Receiver Performance Model 

In one common situation, the required performance measure value for the receiver is known and the 

objective is to determine the corresponding input power ratios (e.g., S/N and S/I). In this case, the 

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analyst steps through the modules in Figure 2-4 in right-to-left order, although part of the RF/IF 

calculation must be done first. The analysis steps for this case are specified below: 

1. Determine which modules in Figure 2-4 apply to the receiver. 

2. Determine the nature of the interference at the output of the RF/IF section. Section 3 describes 

how this is determined. 

3. If the interference is intermittent and a simple manual analysis cannot be performed, stop. 

Section 3 suggests alternate analysis techniques for such cases. 

4. Determine the performance criterion at the receiver output. Depending on the receiver, this 

might be a minimum AI or maximum BER value at the demodulator output, a maximum BER 

value at the FEC decoder output, or a minimum S/N value at the source decoder output. 

5. If there is a source decoder, determine the maximum BER at the source decoder input. Use the 

Input BER vs. Output S/N curves in Section 8. (The maximum BER corresponds to the 

minimum S/N value.)

6. If there is a hard-decision FEC decoder, determine the maximum BER at the FEC decoder input. 

Use the Input BER vs. Output BER curves in Section 7. (The maximum input BER corresponds 

to the maximum output BER.) Then go to Step 8. 

7. If there is a soft-decision FEC decoder, perform the operations described in Steps 8 and 9, except 

that you will use the soft-decision curves in Section 7 rather than the digital demodulator curves 

in Section 6. (The soft-decision curves incorporate both demodulator and FEC decoder effects.) 

8. If the interference at the output of the RF/IF section is noise-like (as determined in Step 2), 

determine the minimum S/N value at the demodulator input. In this case, N represents the total 

noise-like signal power, including receiver noise power and the interference power. Use the 

analog demodulator curves in Section 5 or the digital demodulator curves in Section 6. 

9. If the interference at the output of the RF/IF section is not noise-like (as determined in Step 2), 

select a particular value of S/I and determine the corresponding minimum S/N value at the 

demodulator input. In this case, N represents the receiver noise power. Repeat this for several 

values of S/I. Use the analog demodulator curves in Section 5 or the digital demodulator curves 

in Section 6. 

10. If there is a despreader, calculate the processing gain in dB. Section 4 describes the calculation. 

11. Calculate the frequency-dependent rejection (FDR) in dB of the interference in the RF/IF 

section. Section 3 describes the calculation. 

12. Add the processing gain from Step 9 and the FDR from Step 10 to get the total interference 

power loss in dB. 

13. If the interference at the output of the RF/IF section is noise-like (as determined in Step 2), add 

the total interference power loss in dB from Step 11 to the minimum S/N value in dB from Step 

7. 

14. If the interference at the output of the RF/IF section is not noise-like (as determined in Step 2), 

add the total interference power loss in dB from Step 11 to each S/I value in dB from Step 8. 

2-15 
JSC-CR-06-072 

In another common situation, all signal power levels are known and the objective is to determine the 

performance measure value for the receiver. In this case, the analyst steps through the modules in 

Figure 2-4 in left-to-right order. The same procedure is followed, except that Steps 4 through 14 are 

reversed. 

2-16 
 JSC-CR-06-072 

SECTION 3 – RF/IF SECTION 

This section describes the RF/IF section of a receiver and the model that represents it. It presents the 

FDR concept and shows how the FDR can be calculated. Then it specifies changes in the interfering 

signal waveform that may be caused by the RF/IF section. Finally, for cases involving intermittent 

interference that are beyond the scope of this handbook, it briefly describes the types of analysis that 

might be required. 

3.1 DESCRIPTION 

The RF/IF section of a receiver is the first part of the receiver through which a signal from the antenna 

passes. It generally has amplifiers, mixers, filters, and perhaps other components such as automatic gain 

control circuits. The numbers of each component type and their specific characteristics vary from one 

receiver to another. A receiver performance analysis usually focuses on the amplifiers, mixers, and 

filters. 

Typically, there are several amplifiers in the RF/IF section. Each amplifies the total composite signal at 

its input – including the desired, interfering, and noise components. Because the desired and interfering 

signals experience the same gain in an amplifier, the S/I does not change. However, the RF/IF 

amplifiers are the primary source of receiver noise. The noise from these amplifiers passes through all 

subsequent amplifiers and filters in the RF/IF section. The power of the resultant noise signal is 

referenced to the receiver input, which is equivalent to pretending that the noise signal enters the 

receiver from the antenna and experiences the full gain of all the RF/IF amplifiers. Thus, the S/N is also 

unchanged when the composite signal (including receiver noise) passes through the RF/IF amplifiers. A 

typical noise power calculation is presented in Section 9.2.7. 

If a very strong interfering signal enters a receiver amplifier, it may cause nonlinear effects such as 

desensitization to occur (Section 2.4.3.3). This may happen, for example, when the interfering 

transmitter antenna is close to the receiving antenna and there is very little propagation loss to reduce the 

interfering signal strength. Nonlinear effects occur primarily in the RF/IF section, but they are beyond 

the scope of this handbook. It is assumed that the interfering signal power is sufficiently low that 

nonlinear effects are negligible. With this assumption, and given the fact that the amplifiers do not 

change the S/I or S/N, it is unnecessary to include amplifier effects in the RF/IF section model. 

At the receiver input, the desired signal carrier frequency is tunable. The mixers convert that tunable 

frequency to a fixed frequency. This conversion typically occurs in two or three stages, with a different 

frequency at each stage. The mixer also converts the carrier frequency of the interfering signal in the 

3-1 
JSC-CR-06-072 

same way. Therefore, the frequency separation between the desired and interfering signals is not 

changed by the mixing process, and the filter attenuation of the interfering signal is unaffected by 

mixing. Assuming again that nonlinear effects are negligible, it is also unnecessary to include mixer 

effects in the RF/IF section model. 

Typically, each RF/IF stage contains bandpass filtering to reduce or eliminate unwanted signals. A 

given bandpass filter is centered on a specific frequency. An interfering signal at a given frequency 

separation from this center frequency will generally experience some attenuation. The attenuation vs. 

frequency separation function is the selectivity of the filter. Although different filters in the RF/IF 

section have different center frequencies, the filter selectivities can be aligned and added (in dB) to form

one composite function. Thus the RF/IF section model is simply a model of a single composite filter. 

The selectivity of that filter is known as the receiver selectivity. 

The RF/IF filter produces two effects that significantly impact performance. The first effect is that 

interfering signals may be attenuated by the filter. The FDR is the total attenuation integrated across the 

interfering signal spectrum (Section 3.2). Viewed as a transfer function, the FDR relates the interference 

power at the RF/IF input to the interference power at the RF/IF output. 

The second effect is that the shape of the interfering signal waveform may be changed by the RF/IF

filter (Section 3.3). For example, the pulses in a pulsed radar signal may be smeared together so that the 

resulting signal is no longer pulsed. These waveform changes may result in corresponding changes in 

the receiver performance measure. A specification of the output waveform characteristics as a function 

of the input waveform characteristics can be viewed as a (non-mathematical) transfer function for the 

RF/IF section, because it relates the module outputs to the module inputs. 

3.2 FDR 

The FDR is the total filter loss experienced by an interfering signal in all the RF/IF stages in a receiver. 

The FDR, expressed as a pure ratio rather than in dB, is the ratio of the input interference power to the 

output (filtered) interference power. It is given by: 

∫

∫ ∞

∞

− −

−

=

0

max

0

max

( ) ( )

( )

s s f f r f f df

s s f f df

FDR

T R

T

(3-1) 

3-2 
 JSC-CR-06-072 

where f = absolute frequency, in MHz 

 fT = transmitter tuned frequency, in MHz

 fR = receiver tuned frequency, in MHz 

 smax = maximum power spectral density of the interfering signal at the receiver input, in 

W/MHz

 s(f - fT) = power spectral density of the interfering signal at the receiver input relative to the 

maximum value (unitless ratio) 

 r(f – fR) = receiver selectivity (unitless ratio) 

The integral in the numerator of Equation 3-1 is the power at the receiver input. The integral in the 

denominator of Equation 3-1 is the power at the output of the receiver RF/IF filters. These integrals can 

be evaluated numerically, but Equation 3-1 is often approximated by simpler equations, as shown below. 

The interfering signal is assumed to be on-tune with the receiver if: 





 



 ∆ ≤

2 , 2

Max BT BR f

(3-2) 

where ∆f = frequency separation between the interfering signal and the desired signal, in 

MHz 

Max(a,b) = function that takes the maximum of a and b

 BT = transmitter 3-dB bandwidth, in MHz

 BR = receiver 3-dB bandwidth, in MHz 

In a common approximate model for the transmitter spectrum, it is assumed that s(f - fT) = 1 within BT /2 

of the transmitter tuned frequency and falls quickly to zero outside that band. Thus the numerator of 

Equation 3-1 is approximately equal to smax BT . Similarly for the receiver, it is assumed that r(f - fR) = 1 

within BR /2 of the receiver tuned frequency and falls quickly to zero outside that band. 

For an on-tune signal, the FDR is approximately: 

T R

R

T

R

T

T R

T

T

B B

B

B

s B

s B

B B

s B

s B

FDR

= = >

= = ≤

 if 

1 if 

max

max

max

max

(3-3) 

3-3 
JSC-CR-06-072 

For an off-tune signal, the FDR is approximately: 

( ) ( )

1

( ) ( ) max max

max

R T T R

T

R

R R T T T R

T

s f f r f f B

B

s B s f f s B r f f

s B

FDR

− + − 













 =

− + − =

(3-4) 

Equation 3-4 is obtained by expressing the integral in the denominator of Equation 3-1 as the sum of two 

integrals, one over the receiver region ( f ≈ fR) and one over the transmitter region ( f ≈ fT). It is then 

assumed that s( f - fT) is approximately constant in the receiver region and r( f – fR) is approximately 

constant in the transmitter region. 

Converting to dB, the FDR is given by: 

FDRdB = 0 if on-tune and BT ≤ BR















 =

R

T

B

B FDRdB 10log if on-tune and BT > BR





 



 − + − 













 = −10log ( ) ( ) dB R T T R

T

R s f f r f f B

B FDR if off-tune 

(3-5) 

As an example, suppose that an off-tune interfering signal has a 10-kHz bandwidth and that the receiver 

has a 4-kHz bandwidth. Also suppose that the transmitter spectrum is 20 dB down at the receiver 

frequency and the receiver selectivity is 30 dB down at the transmitter frequency. Then the FDR is 

approximately: 

10 10 23 dB

0.010

0.004 10log

10log ( ) ( )

20 /10 30 /10

dB

=



 



  + 

 



 = −





 



 − + − 













 = −

− −

R T T R

T

R s f f r f f B

B FDR

Expressing the FDR as a transfer function, the input S/I and output S/I are related as follows: 

3-4 
 JSC-CR-06-072 

dB

out in

FDR

I

S

I

S  + 

 



  =  





(3-6) 

3.3 CHANGES TO INTERFERENCE WAVEFORM 

The RF/IF composite filter may change the shape of the interfering signal waveform. A specification of 

the output waveform characteristics as a function of the input waveform characteristics is therefore an 

additional transfer function for the RF/IF section. There are three cases of interest: (1) when either the 

interfering transmitter or the receiver is a frequency hopper, (2) when the interfering signal is pulsed 

(either unmodulated or swept-frequency), and (3) when the interfering signal is digital. 

3.3.1 Frequency Hoppers 

3.3.1.1 Fixed-Frequency Receiver 

Consider the case in which a frequency-hopping transmitter interferes with a fixed-frequency receiver. 

The interfering signal typically hops from one carrier frequency to another at some fixed hop rate (e.g., 

100 hops/s). The dwell time is the duration of transmission at any particular frequency. For a small time 

interval when the transmitter is switching frequencies, the signal is turned off at the transmitter. This 

off-time is usually sufficiently small that the dwell time is approximately the reciprocal of the hop rate. 

When the interfering signal frequency changes, the frequency separation between the interfering signal 

and the receiver changes. As a result, the FDR generally changes. Therefore, the interfering signal 

power at the RF/IF output may vary significantly from one hop to the next. In this case, an input signal 

that is essentially continuous (except for a small off-time) is converted into an intermittent signal that 

has large amplitude variations. 

3.3.1.2 Frequency-Hopping Receiver 

Consider the case in which a narrowband fixed-frequency transmitter interferes with a frequencyhopping receiver. When the desired signal frequency changes, the frequency separation between the 

interfering signal and the receiver changes, and the FDR changes. Therefore, the interfering signal 

power at the RF/IF output may vary significantly from one hop to the next. Just as in the previous case, 

the RF/IF section converts a continuous input signal into an intermittent signal. 

If both the interfering transmitter and the receiver are frequency hoppers, essentially the same

phenomenon occurs. The only difference is that the interfering signal power at the RF/IF output will 

3-5 
JSC-CR-06-072 

change more frequently, because the FDR generally changes when either the interfering signal or the 

receiver changes frequencies. 

Because the intermittent nature of the output interference is caused by fluctuating FDR values, there are 

cases involving frequency hoppers that don’t result in intermittent output signals. In particular, a direct 

sequence interfering signal (Section 4) may have a bandwidth that covers the entire range of frequencies 

in the receiver hopset. In that case, the FDR does not change when the receiver hops because every 

receiver frequency is on-tune with some portion of the interfering signal band. Therefore, the output 

signal is not intermittent.

3.3.2 Pulsed Interfering Signals 

The RF/IF filters in a receiver may distort a pulsed interfering signal in such a way that the pulsewidth 

of the signal changes. The pulses may even be smeared together so that the resulting signal is no longer 

pulsed. In general, the type of distortion depends on whether the pulse is on-tune or off-tune. A pulse is 

on-tune if Equation 3-2 is satisfied. Although the spectrum of a pulsed signal consists of discrete lines, 

the signal bandwidth (which appears in Equation 3-2) is normally based on the envelope of the 

spectrum.

An important characteristic of the RF/IF composite filter is its impulse response, which is the reciprocal 

of the filter bandwidth (1/BR). If BR is in MHz, then 1/BR is in μs. It characterizes the ability of the filter 

to resolve waveform events of short duration. For example, if 1/BR is much less than the pulsewidth τ of 

a single on-tune input pulse, then the filter is able to produce an undistorted output pulse of the same

duration τ. However, if 1/BR is greater than τ, then the output pulsewidth is 1/BR (because 1/BR is the 

smallest resolvable pulsewidth). In this case, the energy of the input pulse is spread out to form a longer 

pulse of lower amplitude. 

Table 3-1 specifies the signal waveform at the RF/IF output when the input interfering signal is pulsed. 

There are nine different cases shown. In all cases, the input signal is pulsed with a pulsewidth τ and a 

pulse repetition interval (PRI), which is the time between the leading edges of two successive pulses. In 

some cases, the frequency of each pulse is linearly swept through a range of frequencies ∆fc. The 

resulting signals are called linear frequency modulated or chirp signals. 

In Table 3-1, “NA” means that the item is not applicable. In Cases 2 through 8, the output waveform is 

a pulse train with the same PRI as the input waveform. In Case 9, the output waveform is a train of 

pulse pairs; the two pulses within a pair are separated by τ and successive pairs are separated by the 

same PRI as the input waveform. 

3-6 
 JSC-CR-06-072 

Table 3-1. RF/IF Output Waveform for Various Interfering Input Signals 

Case ∆f Modulation

Bandwidth 

Condition

Resolution 

Condition Output Waveform

1 On-tune or 

off-tune 

Unmodulated

or chirp 

NA ≥ PRI > τ

BR

1

Continuous CW-like signal

2 On-tune Unmodulated NA > > τ

BR

PRI 1 One pulse per PRI with 

pulsewidth 1/BR

3 On-tune Unmodulated NA BR

PRI 1> τ ≥ One pulse per PRI with 

pulsewidth τ

4 On-tune Chirp BT > BR τ

T

R

R B

B

B

PRI > >

1 One pulse per PRI with 

pulsewidth 1/BR

5 On-tune Chirp BT > BR

T R

R

B B

B PRI 1

> τ ≥ One pulse per PRI with 

pulsewidth (BR /BT)τ

6 On-tune Chirp BR ≥ BT > > τ

BR

PRI 1 One pulse per PRI with 

pulsewidth 1/BR

7 On-tune Chirp BR ≥ BT BR

PRI 1> τ ≥ One pulse per PRI with 

pulsewidth τ

8 Off-tune

Unmodulated

or chirp 

NA > > τ

BR

PRI 1 One pulse per PRI with 

pulsewidth 1/BR

9 Off-tune

Unmodulated

or chirp 

NA 

BR

PRI 1> τ ≥ Two pulses per PRI, separated 

by τ, each with pulsewidth 1/BR

3.3.2.1 Unresolvable Pulses 

In Case 1, the impulse response 1/BR is greater than the time between pulses (PRI). This case includes 

on-tune or off-tune signals that may be chirp or unmodulated. A single pulse by itself would be 

stretched to the duration 1/BR. However, because this duration is greater than the time between pulses, 

the stretched pulses in a pulse train overlap and form a continuous waveform. The exact nature of this 

waveform depends on where the spectral lines lie relative to the receiver passband. In the worst case, 

there is one spectral line within the receiver passband. In this case, the output waveform is typically 

CW-like. 

3-7 
JSC-CR-06-072 

3.3.2.2 On-tune and Unmodulated Pulses 

In Cases 2 and 3, the input interfering signal is an on-tune and unmodulated pulsed signal. In Case 2, 

the impulse response is less than the PRI but greater than the input pulsewidth. Each pulse is stretched 

to the duration 1/BR. Thus, the output waveform has one pulse of pulsewidth 1/BR for each input pulse. 

In Case 3, the impulse response is less than the input pulsewidth, so the filter is able to produce an 

undistorted output pulse of the same duration. Thus, the output waveform has one pulse of pulsewidth τ

for each input pulse. 

3.3.2.3 On-tune Shortened Chirp Pulses 

In Cases 4 and 5, the input interfering signal is an on-tune chirp signal. For this type of signal, there is 

one additional factor to consider. If the frequency of the pulse is swept through a range of frequencies 

∆fc that exceeds the IF bandwidth, then the pulse will be attenuated at any frequencies outside the IF

bandwidth. This shortens the pulsewidth by the factor BR /∆fc. Because the transmitter bandwidth BT is 

nominally equal to ∆fc, the shortening factor is BR /BT. The bandwidth condition BT > BR specifies that 

∆fc does exceed the IF bandwidth. Cases 4 and 5 are the same as Cases 2 and 3, except that the 

shortened pulsewidth [(BR /BT) τ] replaces the pulsewidth τ. 

3.3.2.4 On-tune Unshortened Chirp Pulses 

Cases 6 and 7 are the same as Cases 4 and 5, except that the bandwidth condition BR ≥ BT specifies that 

∆fc does not exceed the IF bandwidth. Since the pulse does not sweep to frequencies outside the IF 

bandwidth, the pulse shortening effect does not occur. Therefore, Cases 6 and 7 are equivalent to Cases 

2 and 3, respectively. 

3.3.2.5 Off-tune Pulses 

In Cases 8 and 9, the input interfering signal is an off-tune pulsed signal. For an off-tune pulse, whether 

it is an ordinary or a chirp pulse, the filter produces an impulse response at each edge of the pulse that 

experiences less attenuation (FDR) than the center of the pulse. If 1/BR is less than τ (Case 9), then the 

resulting output waveform is a pair of pulses of width 1/BR, separated by the original pulsewidth. If 1/BR

is greater than τ (Case 8), then the two edge responses overlap to form one pulse of width 1/BR. 

3.3.2.6 Analysis Considerations 

Case 1 in Table 3-1 is the only case in which the RF/IF section converts a pulsed signal into a signal that 

is not intermittent. In this case, the methods of this handbook can be applied. Because the RF/IF output 

3-8 
 JSC-CR-06-072 

signal is not pulsed, the average power of the signal (averaged over a complete pulse repetition interval), 

rather than the peak power (power during the pulse), should be used in the analysis. These are related as 

follows: 





 



 = +

PRI

P P τ

avg peak 10 log

(3-7) 

where Pavg = average power of the signal, in dBm 

 Ppeak = peak power of the signal, in dBm

τ = pulsewidth, in μs 

 PRI = pulse repetition interval, in μs 

The output signal in Cases 2 through 8 is pulsed, and so the methods of this handbook cannot be applied 

to those cases. They have been included in Table 3-1 for those who may wish to apply other methods. 

Section 3.4 gives a brief introduction to the types of analysis that may be appropriate. In such an 

analysis, it is necessary to include the fact that the peak power of the output pulse may differ from the 

peak power of the input pulse, not only because of FDR but also because the energy is spread across a 

different pulsewidth. These are related as follows: 

P P FDR − a 













 = − +

out

out in dB 10 log

τ

τ

(3-8) 

where Pout = peak power of the output pulse, in dBm 

 Pin = peak power of the input pulse, in dBm 

FDR dB = frequency dependent rejection, in dB (Section 3.2) 

τ = input pulsewidth, in μs 

τout = output pulsewidth, in μs 

a = 3 dB for Case 9 (each output pulse has half the power) 

= 0 dB for Cases 2 through 8 

3.3.3 Digital Interfering Signals 

A digital interfering signal is similar in some ways to an unmodulated pulsed signal. Each bit (or chip 

for a spread spectrum transmitter) can be thought of as a pulse, but with no “dead time” between pulses. 

Many of the pulsed signal concepts from Section 3.3.2 are also applicable to digital signals, but most 

3-9 
JSC-CR-06-072 

types of distortion have little effect on receiver performance. However, one type of distortion occurs 

that is significant: when the interfering signal bit duration is less than the impulse response (1/BR) of the 

RF/IF composite filter, then the output interfering signal is a noise-like signal. 

3.4 ANALYSIS WITH INTERMITTENT SIGNALS 

When interfering signals are intermittent, the straightforward analysis method of this handbook cannot 

be used. A complete description of an appropriate analysis method for that case is beyond the scope of 

this handbook, but the types of analysis that might be required will be briefly described. 

As an initial example, consider a digital receiver with no FEC and an interfering signal that has a power 

level I for 2% of the time and negligible power for the other 98%. It could be a pulsed radar signal for 

which the pulsewidth is 2% of the pulse repetition interval. Or it could be a narrowband fixed-frequency 

interfering signal that is in the same channel as one of the 50 hopset frequencies of a frequency-hopping 

receiver. If the receiver is affected by the interference only when it falls in the same channel, then that 

co-channel event is like an interference pulse whose duration is 2% (1/50) of the total time spent 

hopping through the entire hopset. For either case, the BER is given by: 

BER = BERN + 2 BERI +N 0 .98 0.

(3-9) 

where BER = average BER 

 BER N = BER in the presence of noise 

 BER I+N = BER in the presence of both interference and noise 

Alternatively, rather than calculate a BER, the pulse may be called an outage and the statistics of the 

outage (e.g., a 10 ms outage that occurs every 500 ms) may simply be reported. 

Although this simple example can be analyzed with a simple manual technique, many cases are too 

complex to analyze manually. For example, there may be multiple radars with different pulsewidths and 

power levels simultaneously interfering with the receiver. Or the narrowband fixed-frequency signal 

interfering with the frequency-hopping receiver could have many amplitude levels at the RF/IF output 

(rather than just two levels) because of changes in the FDR as the receiver hops. Equation 3-9 can be 

generalized as follows: 

= ∑

j

BER p jBERj

(3-10) 

3-10 
JSC-CR-06-072 

where BER = average BER 

 p j = probability that the jth interference level occurs 

 BER j = BER when the interference is at the jth level 

It may be difficult to determine the statistics (e.g., values of pj) required to use Equation 3-10. In 

addition, if the receiver has FEC then Equation 3-10 may not be valid. In that case, the relative duration 

of an interference event and the FEC codeword may be important. (A brief discussion of FEC with 

intermittent signals is given in Section 7.4.) The best approach for analyzing these complex cases is the 

simulation approach (Section 2.4.4). 

Note that the desired signal may also be intermittent. For example, if a frequency-hopping receiver is 

subject to frequency-selective multipath fading, then the desired signal power will fluctuate (fade) as the 

receiver hops. 

3.5 SUMMARY 

There are two transfer functions associated with the RF/IF section of a receiver. The first is the FDR,

which relates the S/I at the RF/IF input to the S/I at the RF/IF output according to Equation 3-6. The 

FDR is defined by Equation 3-1 and can be approximated in dB by Equation 3-5. The second is a nonmathematical transfer function that relates the interference waveform characteristics at the RF/IF input 

to the interference waveform characteristics at the RF/IF output. It is specified for frequency hopping, 

pulsed, and digital signals in Sections 3.3.1, 3.3.2, and 3.3.3, respectively. 

3-11/3-12 
 JSC-CR-06-072 

SECTION 4 – DESPREADER 

This section describes spread spectrum processing and characterizes the processing gain associated with 

the despreader in a spread spectrum system. Because spread spectrum is frequently associated with 

multiple access, some simple multiple-access equations are also presented. 

4.1 DESCRIPTION 

In a transmitter, spreading is the process of multiplying the narrowband information signal by a 

wideband code waveform to dramatically increase the bandwidth of the signal. In the receiver, 

despreading multiplies the wideband desired signal by the same code waveform to recover the original 

narrowband signal. A spread spectrum system is a system that employs spreading and despreading. An 

RF system may employ spread-spectrum technology for several reasons: 

•

•

•

•

•

•

To enable multiple-access communications with many users utilizing the same portion of the RF 

spectrum at the same time without unacceptable mutual interference 

To suppress narrowband interference 

To resist multipath fading

The operation of a spread-spectrum system is based on a pseudo-noise (PN) spreading code set, which is 

known by both the transmitter and receiver. A PN code set is a collection of binary sequences (called 

keys, spreading codes, or simply codes) with the following properties: 

The autocorrelation of each code is very small. 

The cross-correlation of multiple distinct codes is very small.

The RF waveform representing each code appears to be noise-like. 

The desired signal is spread with a particular PN code waveform. When the despreader multiplies the 

desired signal by that same PN code waveform, it removes the code. As a result, the original 

narrowband desired signal is recovered. This despreading works because the receiver-generated PN

code waveform is synchronized with the PN code waveform on the desired signal and is, therefore, 

perfectly correlated. After the despreader, the narrowband desired signal passes through a narrow 

bandpass filter. 

If the receiver is part of a multiple-access system and the received signal includes an interfering signal 

from a different user in the system, then that signal is also multiplied by the receiver’s PN code 

waveform. However, the PN code on the interfering signal does not match the receiver’s PN code. 

4-1 
JSC-CR-06-072 

Because the two codes are uncorrelated, the interfering signal is not despread. When the wideband 

interfering signal at the despreader output then passes through the narrow bandpass filter, much of it is 

attenuated. This attenuation is what allows a spread spectrum multiple-access receiver to distinguish 

and extract the desired signal from the composite signal that includes signals from multiple users. 

The same mechanism in a spread spectrum receiver suppresses narrowband interference. When the 

despreader multiplies the narrowband interfering signal by the receiver-generated PN code waveform, 

the interfering signal is spread (becoming a wideband signal). When that wideband signal then passes 

through the narrow bandpass filter, much of it is attenuated. 

In both of these cases, despreading (followed by narrowband filtering) can therefore produce a dramatic 

improvement in the S/I. This improvement is called the processing gain. The types of spreading are 

direct sequence, frequency hopping, and time hopping. 

Spread-spectrum systems resist fading. In a frequency-selective multipath fading environment, typically 

only a small portion of the large spread-spectrum bandwidth experiences fading. The degradation 

normally caused by propagation delays is minimized by the despreader. 

4.2 DIRECT SEQUENCE 

In a direct sequence system, the code waveform is usually a long pseudorandom sequence of shortduration bits known as chips. The transmitter spreads a digital signal by multiplying the signal by the 

code waveform. The coded waveform is typically transmitted and received using a binary digital 

modulation scheme, such as phase-shift keying (PSK) or differential PSK (DPSK). The spreading can 

occur before, during, or after the modulation process. 

As an example of one common implementation, consider a case with 100 chips per bit. The information 

bit sequence is combined with FEC bits to produce the coded bit waveform b(t), which is a series of

rectangular pulses. Each pulse has an amplitude ±1. The FEC-coded bit rate is Rb in bits/s. The 

spreading sequence is represented by the PN code waveform c(t), which is also a series of rectangular 

pulses of amplitude ±1. The chip rate of the PN code is Rc = 100 Rb. In the transmitter, the spreader 

multiplies the two waveforms together to get the spread baseband waveform b(t) c(t). That waveform

then modulates a carrier to get a PSK signal. In the receiver, the received PSK signal is demodulated to 

recover the baseband waveform b(t) c(t). The despreader then synchronizes its local PN code waveform

c(t) with the input signal and multiplies the two waveforms to get [b(t) c(t)][c(t)] = b(t) c

2

(t) = b(t). 

Thus, in the absence of interference and noise, the original FEC-coded bit waveform b(t) is recovered. 

If interference and noise are not negligible, then the received bit sequence will contain bit errors and the 

FEC decoder will attempt to correct those errors.

4-2 
 JSC-CR-06-072 

4.2.1 Processing Gain 

The RF bandwidth Bc associated with the desired signal waveform before despreading is much larger 

than the bandwidth Bb associated with the desired signal waveform after despreading. The bandpass 

filter following the despreader has a nominal bandwidth Bb. 

When the spread-spectrum desired signal is accompanied by narrowband interference, the despreader 

expands the interfering signal bandwidth from BI to approximately Bc. The bandpass filter then 

attenuates the interference by the factor Bc/Bb. This factor, expressed in dB, is the spread-spectrum 

processing gain: 

( ) M

R

R

B

B G

b

c

b

c

SS 10 log 10 log = 10 log 













 ≈













 =

(4-1) 

where GSS = spread-spectrum processing gain, in dB 

 Bc = 3-dB bandwidth of the desired signal before despreading, in MHz 

 Bb = 3-dB bandwidth of the desired signal after despreading, in MHz 

 Rc = chip rate of despreader, in chips/s 

 Rb = bit rate at despreader output, in bits/s 

 M = number of chips per bit 

If the interference is wideband with BI > Bc, then the RF/IF section of the receiver (Section 3) reduces 

the interfering signal bandwidth from BI to approximately Bc. That bandwidth reduction is accompanied 

by an attenuation, which is included in the FDR (Section 3.2). When the despreader multiplies the 

interfering signal by the PN code waveform, the interfering signal bandwidth remains approximately 

equal to Bc. The bandpass filter then attenuates the interference by the factor Bc/Bb. Therefore, Equation 

4-1 also applies to the wideband interference case. Note that the FDR accounts for filter attenuation in 

the RF/IF section, whereas the processing gain accounts for filter attenuation in the bandpass filter that 

follows the despreader. 

Multipath fading, which is a form of self-interference, can be modeled as narrowband interference to a 

direct-sequence system. Typically, a fade margin is planned for a communications link in which fading 

is anticipated. This fade margin can be reduced (by an amount less than or equal to the coding gain) 

when the system uses direct-sequence spread-spectrum communication. 

4-3 
JSC-CR-06-072 

4.2.2 Multiple-Access Interference 

Since direct-sequence spread-spectrum systems are typically designed to support multiple-access 

communications within a single frequency band, it is frequently necessary to consider the effects of the 

multiple-access interference as well as noise. If it is assumed that there are K simultaneous users, then 

there will be one desired signal and K-1 undesired signals. If it is further assumed that the K signals are 

of equal power at the PSK receiver input, then the BER can be approximated as:4-1





























+ − =

Eb

N

M

K

BER Q

3 2

1

1

o

(4-2) 

where K = total number of simultaneous users 

 M = number of chips per bit 

 No = noise power density, in W/Hz 

 Eb = energy per bit, in J (or W/Hz) 

The variables Eb and No are discussed in Section 6. The function Q(X) is defined as follows:4-2

∫

∞

− =

X

y Q X e dy / 2 2

2

1 ( )

π

(4-3) 

For a single user the system is limited by noise: 















 =

o

2

N

E

BER Q b

For a large number of users, the system is limited by the multiple-access interference: 

 

4-1 Theodore S. Rappaport, Wireless Communications Principles and Practice, 2nd ed., Upper Saddle River, NJ: 

Prentice Hall PTR, 2002. 4-2 John G. Proakis, Digital Communications, 3rd ed., McGraw-Hill Series in Electrical Engineering, 1995. 

4-4 
 JSC-CR-06-072 

















− = 1

3

K

M BER Q

These equations assume that signals are received from all users with equal power. If this assumption is 

not valid, then other more complex equations may be used. To use these complex equations, the analyst 

must specify statistics describing the distribution of received signal power from all of the users. The 

only way to obtain such statistics is by field measurements. If such measurements are made, and it is 

determined that the variation of signal power is limiting the system capacity, the typical response is to 

reduce the variation. This can be done, for example, by reducing the coverage area or by inserting 

repeaters into the coverage area. 

4.3 FREQUENCY HOPPING 

In a frequency-hopping system, the desired signal is typically a digital signal that hops from one carrier 

frequency to another at some fixed hop rate. A slow hopper is one that transmits several bits during

each hop. A fast hopper is one that hops several times during each bit. The system hopset is the set of

possible frequencies. The transmitter uses a PN code to determine the sequence of frequencies from the 

hopset that will be used in a transmission. Typically, the PN code waveform is a long pseudorandom 

sequence of discrete voltages that control the output frequency of an oscillator. The receiver uses the 

same PN code to convert the hopping desired signal to a fixed-frequency signal. 

4.3.1 Processing Gain 

Each of the frequencies in a hopset is the center frequency of what is called an instantaneous channel. 

Each instantaneous channel has an instantaneous bandwidth Bb. Typically, binary frequency-shift 

keying (FSK) modulation is used, so there are two signalling frequencies within each channel. 

Assuming that there are M contiguous carrier frequencies and channels, the total system bandwidth is BS

= MBb. (However, a given channel is sometimes removed from a hopset to avoid interference in the 

environment.) 

In a frequency-hopping receiver, a hopping local oscillator signal mixes with the desired signal. This 

process eliminates the desired signal hopping and converts a non-hopping interfering signal into a 

hopping signal. As in the case of a direct sequence receiver, this shrinks the desired signal bandwidth 

and spreads the interfering signal spectrum. A non-hopping narrowband filter then filters out most of 

the interfering signal. This mixing and narrowband filtering normally occurs in the RF/IF section, so the 

resulting attenuation is already included in the FDR (Section 3.2). Therefore, although the concept of 

“processing gain” can be applied to a frequency-hopping receiver, Equation 4-1 is not used. 

4-5 
JSC-CR-06-072 

As discussed in Section 3.3.1, the interfering signal at the RF/IF output in a frequency-hopping receiver 

is usually intermittent, whether the hopper is slow or fast. Thus, the straightforward analysis method of 

this handbook cannot be used. Section 3.4 discusses analysis options for such cases. 

4.3.2 Multiple-Access Interference 

In a frequency-hopping multiple-access system, mutual interference is minimized by designing the 

hopping sequences so that the probability of two users hopping to the same channel at the same time is 

small. If two users hop to the same channel at the same time, the event is called a hit. For a slow 

hopper, it may be assumed that during a hit the probability of bit error is 0.5. Adapting Equation 3-10 to 

this case, the BER is given by: 

miss miss miss

hit hit miss miss

(1 p )(1/ 2) p BER

p BER p BER

BER p BER

j

j j

= − +

= +

= ∑

(4-4) 

where phit = probability that a bit is in a hit 

 pmiss = probability that a bit is in a miss (not a hit) 

 BERhit = BER during a hit 

 BERmiss = BER during a miss 

The probability of a hit is related to the number of simultaneous users, and the number of instantaneous 

carrier frequencies in the hopset. From the point of view of a single receiver, it cannot be assumed that 

all the signals hop synchronously. Even if there were a master system clock, variations in propagation 

delay would result in varying degrees of offset. In this case, the probability that a bit is not in a hit is 

(Reference 4-1): 

1

miss

1 1 1 1

−





 



 



 



 = − +

K

M L p

(4-5) 

4-6 
 JSC-CR-06-072 

where M = number of frequencies in the hopset 

 L = number of bits per hop 

 K = number of simultaneous users 

4.4 TIME HOPPING 

In a conventional digital system, the bit duration is the reciprocal of the bit rate. For example, if the bit 

rate is 1000 bps then the bit duration is 1 ms. In a time-hopping system, the conventional bit interval 

(reciprocal of the bit rate) is conceptually subdivided into small subintervals. The bit is transmitted in 

one of these subintervals. To point out the similarity to a direct sequence system, the small bit can be 

referred to as a chip. The position of the chip within the conventional bit interval is dictated by the PN

code waveform. The frequency does not hop. Spreading occurs because the chip duration is much less 

than the conventional bit duration. 

In a time-hopping receiver, the PN code waveform dictates where in the hopping interval the pulse is 

located. No energy is collected outside of that pulse window, so most of the interfering signal power is 

not collected. Equation 4-1 can be used to calculate the processing gain. 

For example, consider a system with no FEC for which the conventional bit duration is 100 times the 

chip duration. If the bit rate is 1000 bps, then the conventional bit duration is 1 ms and the chip duration 

is 0.01 ms. In the transmitter, the information chip is placed in the n

th chip location, where n is a

pseudorandom integer between 1 and 100 specified by the PN code. In the receiver, the despreader uses 

the same PN code to determine the value of n. It then converts the chip waveform back into the 

conventional bit waveform. The number of chips per bit (ratio of conventional bit duration to chip 

duration) is 100, so the processing gain is 10 log(100) = 20 dB. 

4.5 SUMMARY 

The transfer function associated with the despreader is the spread-spectrum processing gain, which 

relates the S/I at the despreader input to the S/I at the despreader output. The spread-spectrum

processing gain is given by Equation 4-1. The S/I is increased by that gain in a direct sequence or time

hopping receiver. 

4-7/4-8 
 JSC-CR-06-072 

SECTION 5 – ANALOG DEMODULATOR 

This section describes the demodulator model for amplitude modulation (AM) voice, frequency 

modulation (FM) voice, and broadcast television receivers. Subjective performance measures of 

intelligibility for voice systems are discussed. Transfer functions that relate output S/N to input S/N are 

presented. Curves that specify intelligibility measures as a function of S/N and S/I are also displayed. 

5.1 INTRODUCTION 

The purpose of the analog demodulator is to remove the RF carrier and restore the baseband signal that 

was used to modulate the carrier. There are two common types of analog modulation – AM and FM – 

each with advantages and disadvantages pertinent to RF communications. 

There are several properties of analog RF systems that affect the system performance: 

Bandwidth. AM systems require less bandwidth than FM systems. A typical AM system requires an 

RF channel that is approximately twice the width of the baseband signal spectrum. Since an analog

speech signal has a baseband width of less than 4 kHz, an AM system can accommodate many channels. 

Single-sideband AM systems require even less bandwidth – approximately the width of the baseband 

spectrum. FM systems require several times the bandwidth of the baseband spectrum.

Susceptibility to Noise and Interference. FM systems are generally less susceptible to noise, and are 

particularly less susceptible to intermittent interference and fading than AM systems. FM systems have 

a property that allows this immunity to be increased by increasing the RF bandwidth. AM systems do 

not have this property, leaving an increase in transmitter power as the principle means for overcoming 

noise and interference. 

Threshold. Analog systems have a certain threshold, which is a received-signal power below which no 

usable output is obtained. AM systems can often operate at lower sensitivities than FM systems. FM 

systems, on the other hand, exhibit the so-called capture effect, which controls the way in which the 

output performance relates to increasing received-signal power. The capture effect allows the 

performance of the system, once the sensitivity threshold has been reached, to increase. 

5.2 VOICE PERFORMANCE MEASURES 

The performance of RF communications receivers is typically quantified in terms of S/N, or some

similar ratio of desired-signal power to undesired-signal power. This is an objective measure of 

performance, in the sense that it can be measured with RF test equipment. 

5-1 
JSC-CR-06-072 

Objective measures of performance do not directly address properties such as fidelity or intelligibility, 

since these properties depend on human perception. Nevertheless, it is these properties that are 

ultimately of interest to users of RF communications systems. Consequently, there have been attempts 

to develop measures of performance that are more closely aligned with human perception. 

5.2.1 Articulation Index 

One of the most important applications of RF communications technology is the transmission of speech. 

The spectrum of the speech signal extends from roughly 200 Hz to 2500 Hz. The most basic and 

fundamental property that can be applied to a speech signal is intelligibility – that is, the ability of the 

listener to understand the speaker. 

Certain parts of the speech spectrum are more sensitive to noise and interference than others. For this 

reason, a measure of performance was developed that assigns varying weights to undesired signal 

power, based on the part of the speech spectrum that the signal occupies. This measure of performance, 

called the AI, is based on empirical data that shows the correlation between the undesired signal 

spectrum and the intelligibility of the speech. Although the basis of the AI is empirical, the definition of 

AI is not. The band from 200 Hz to 6100 Hz is divided into 14 contiguous sub-bands of unequal width. 

The S/I in each sub-band is calculated and normalized to give a value between 0 and 1. The average of 

these 14 normalized values is the AI. It is effectively a frequency-weighted average because the subbands are of unequal width. 

In practice, AI is used to characterize the effect of undesired signals on transmitted speech by assigning 

a qualitative descriptor to a range of AI values. A typical assignment might be: 

1.0 > AI > 0.9 Good intelligibility 

0.9 > AI > 0.7 Marginal intelligibility 

0.7 > AI Unacceptably poor intelligibility 

Because the AI model is based on the long-term average speech and interfering signal spectrum, it 

cannot be used when either signal is intermittent.

5.2.2 Articulation Score 

The ultimate test of system performance is to actually measure the intelligibility of transmitted speech 

by empirical methods. The experiments are designed to facilitate the comparison of a spoken message 

with the same message after it has been transmitted and received. The results are quantified by counting 

5-2 
 JSC-CR-06-072 

the number of correct vs. incorrect words in the received message as reported by test listeners. The 

percentage of correct words is called the AS. The AS is thus the most direct measure of system

intelligibility. 

When designing AS experiments, it is important – but extremely difficult – to minimize the contribution 

of other variables that affect intelligibility. These variables include: 

Age, education, and regional accent of speakers and listeners. AS experiments have been performed 

primarily on college campuses, using college students as listeners. This introduces the possibility that 

the uniform level of education (typically first or second year undergraduate) will bias the results. 

Furthermore, unless a number of colleges in different locations are involved (historically this is not the 

case), the local accents and speech patterns will bias the results. 

Type of message content. The message may be designed to communicate information in small, 

discrete, uncorrelated units (for example numbers or lists of words). The units may be larger, yet remain 

discrete and more or less uncorrelated, such as a news broadcast. The message may be highly 

correlated, where there is one central idea, and individual words are not identically important. The AS 

experiments favor the small, uncorrelated message units, but many real-world communications are very 

different. 

Style of delivery. The message may be spoken in casual, conversational tones, or it may be delivered 

with urgency. The focus, concentration, and ultimately the AS of listeners can be affected by whether 

the message is delivered with a declarative or imperative style. 

Vocabulary. If speakers employ a limited vocabulary which is known to and anticipated by the 

listeners, the AS will be much higher than otherwise. Limited vocabularies and very disciplined 

delivery styles are used for military, emergency response, scientific, and other types of technical 

communications. These systems can often be operated in very noisy RF environments. 

The impact of these variables is such that AS results have very narrow applicability. The degree to 

which a particular system resembles a test system with respect to these variables is the degree to which 

the AS tests will prove useful. There have been attempts to relate AS to AI, since AI is supposed to

factor in frequency-dependent intelligibility. Even so, the relationship between AS and AI varies 

widely, exhibiting strong dependence on the aforementioned variables. 

5-3 
JSC-CR-06-072 

5.3 AM VOICE 

5.3.1 Description 

The waveform of an AM signal is given by: 

v t A [ m t ] ( f t) c 2π c ( ) = 1+ ( ) cos

(5-1) 

where v(t) = waveform magnitude at time t, in V 

 t = time, in s 

 Ac = constant signal amplitude, in V

 m(t) = baseband (modulating) signal, unitless 

 fc = carrier frequency, in Hz 

For AM voice, the modulating signal m(t) is the voice signal to be transmitted. The original voice signal 

is peak-limited and scaled so that the peak amplitude of |m(t)| is not greater than 1. That peak amplitude 

is called the modulation index μ of the AM signal. 

Amplitude modulation results in frequency translation of the baseband spectrum. The translated 

baseband spectrum is referred to as a sideband. Equation (5-1) results in a pair of sidebands – each a 

replica of the baseband spectrum – centered about the carrier frequency fc. There are several variations 

of amplitude modulated systems in which parts of the signal are suppressed. This section considers the 

full AM signal with carrier and two sidebands. The variations are specified in Section 5.4. 

5.3.2 Performance 

Important considerations are the demodulator threshold, sensitivity, and the output S/N. These are 

functions of the input S/N. It should be noted that the AM demodulator can never improve upon the 

input S/N. Ideal AM demodulation would result in an output S/N equal to the input S/N. 

The demodulator needs a certain minimum input S/N to function properly. The AM threshold is a point 

on the curve of output S/N vs. input S/N at which the curve changes dramatically. Above threshold, the 

input vs. output S/N relationship is linear. Below threshold, the output S/N falls off faster than the input 

S/N. The threshold varies somewhat, depending on the type of demodulator circuit, but a reasonable 

approximation is an input S/N of approximately 5 dB. This is so low that, even if the demodulator were 

operating in the linear range, the output would not be usable. 

5-4 
 JSC-CR-06-072 

Sensitivity is the minimum received-signal power required for proper receiver operation in an 

interference-free environment. For analog receivers, “proper operation” is usually specified in terms of

the output S/N. Therefore, a design goal for the RF engineer is to ensure that the required receivedsignal power is available. Sensitivity is usually determined experimentally. 

In some analyses not involving interference, it may be sufficient to simply compare the received-signal 

power with the sensitivity. In other cases, it may be necessary to estimate the actual output S/N. For 

example, there may be a range of 10 to 15 dB in input S/N between the demodulator threshold and the 

sensitivity. If the system is operating in this range, the output will be noticably degraded, but possibly 

still useful. At the other extreme, for critical messages it may be necessary to operate with a higher 

output S/N than that specified in connection with sensitivity. 

5.3.3 Transfer Function 

For an AM demodulator operating above threshold with noise (or noise-like interference), the output 

S/N is related to the input S/N as follows: 

















+ < >

< >  + 

 



  = 

 





1 ( )

( ) 10 log 2

2

out in m t

m t

N

S

N

S

(5-2) 

where (S/N)out = output signal-to-noise power ratio, in dB 

(S/N)in = input signal-to-noise power ratio, in dB 

< ( ) > = time average of the square of m(t), unitless 2

m t

The numerator in the rightmost term of Equation 5-2 is the power of the modulating signal (in 1 ohm)

relative to the carrier power. In a typical application, the root-mean-square (RMS) voltage of the voice 

signal m(t) is 0.3 (relative to the carrier amplitude). Therefore,< = 0.09 and the rightmost term

has the value -11 dB. The output S/N is 11 dB lower than the input S/N because the demodulator 

eliminates the carrier, which contains most of the power of the input signal. 

( ) > 2

m t

If the modulating signal is a sinusoidal tone with modulation index μ (rather than a voice signal), then 

the transfer function is: 

















+

 + 

 



  =  





1 / 2

/ 2 10 log 2

2

out in μ

μ

N

S

N

S

(5-3) 

5-5 
JSC-CR-06-072 

For an AM demodulator operating above threshold with AM voice interference, simulations show the 

transfer function to be: 

8

out in

 − 

 



  =  





I

S

I

S

(5-4) 

In these simulations, the RMS voltage of each voice signal was set to 0.3 (relative to the carrier 

amplitude). 

5.3.4 AI Curves 

Figures 5-1 and 5-2 show AI curves for an AM voice receiver. In these graphs, the term “interference” 

and the variable I refer to non-noise-like interference. Any noise-like interference power is simply 

assumed to be added (in mW) to the noise power. For example, suppose there are two interfering 

signals: an amplitude-shift keying (ASK) signal and a noise-like signal. The S/I parameter is the ratio 

of the desired signal power to the ASK interfering signal power, and the x-axis variable S/N is the ratio 

of the desired signal power to the total noise-like power (including the receiver noise and the noise-like 

interfering signal) at the input of the receiver. 

The curves were generated by time-domain simulation. The IF bandwidth was 8 kHz. The audio 

(baseband) bandwidth was from 300 Hz to 3.5 kHz. The modulation index of the desired signal was 0.3. 

In the last part of the simulation, the signals were transformed to the frequency domain for calculating 

the AI, as described in Section 5.2.1. 

Figure 5-1 shows AI curves for an AM voice receiver with ASK (on-off keying) interference. There are 

six curves displayed. Each curve is a plot of AI vs. S/N. The top curve (labeled “No interference”) 

applies to the case in which there is no ASK interference. The other five curves are for cases with ASK 

interference. Each of those five curves is labeled with the S/I for that curve. As expected, each curve 

shows that the intelligibility (as indicated by the AI) improves as the S/N increases. As also expected, 

for a given S/N the intelligibility improves as the S/I increases. 

In the simulation, the ASK interfering signal had a bit rate of 800 bps, but other simulations showed the 

results to be insensitive to the bit rate. (However, at very high bit rates the interfering signal is noiselike, as specified in Section 3.3.3.) The ASK signal was tuned 500 Hz away from the receiver, but other 

simulations showed the results to be insensitive to variations of off-tuning within the same frequency 

channel. 

5-6 
 JSC-CR-06-072 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30

S/N (dB)

AI

40

No interference S/I = 20 dB

S/I = 15 dB

S/I = 10 dB

S/I = 5 dB

S/I = 0 dB

Figure 5-1. AI vs. (S/N)in Curves for AM Voice Receiver with ASK Interference

5-7 
JSC-CR-06-072 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 4

S/N (dB)

AI

0

No interference

S/I = 20 dB

S/I = 15 dB

S/I = 10 dB

S/I = 5 dB

S/I = 0 dB

Figure 5-2. AI vs. (S/N)in Curves for AM Voice Receiver with FSK Interference 

5-8 
 JSC-CR-06-072 

Figure 5-2 shows AI curves for an AM voice receiver with FSK interference. In the simulation, the FSK 

interfering signal had a bit rate of 50 bps and an off-tuning of 500 Hz, but other simulations showed the 

results to be insensitive to the bit rate and to variations of off-tuning within the same frequency channel. 

The frequency deviation of the FSK signal was ± 200 Hz. 

5.4 SINGLE SIDEBAND VOICE 

5.4.1 Description 

As described in Section 5.3, amplitude modulation results in a pair of sidebands – each a replica of the 

baseband spectrum – centered about the carrier frequency. There are several variations of amplitude 

modulated systems: 

• Double sideband (DSB). This refers to the entire AM signal – the carrier and both sidebands. It 

is addressed in Section 5.3. 

• DSB with suppressed carrier. The two sidebands are transmitted, but not the carrier.

• Single sideband (SSB). Normally this means that only one of the sidebands is transmitted 

without the carrier. 

• SSB with pilot tone. A single sideband is transmitted along with a pilot tone. The pilot tone is 

transmitted to aid in the demodulation process. This technique is especially useful in mobile 

systems, since it makes it possible to eliminate the effects of doppler shifts. The pilot tone may 

be the actual carrier frequency, but it does not have to be. 

• Vestigial sideband. The carrier, one complete sideband, and a small portion of the other 

sideband are transmitted. This technique is used by analog broadcast television. It is addressed 

in Section 5.6. 

5.4.2 Transfer Function 

For systems that do not transmit the RF carrier, the output S/N is the same as the input S/N, assuming 

the system is operating above threshold: 

out in





 



  =  





N

S

N

S

(5-5) 

5-9 
JSC-CR-06-072 

Equation 5-5 applies to SSB receivers and to DSB with suppressed carrier. For SSB systems with a 

pilot tone, the pilot tone may be assumed to be small enough that Equation 5-5 applies. 

5.4.3 AI Curves 

Figures 5-3 and 5-4 show AI curves for a SSB voice receiver. In these graphs, the term “interference” 

and the variable I refer to non-noise-like interference. Any noise-like interference power is simply 

assumed to be added (in mW) to the noise power. For example, suppose there are two interfering 

signals: an ASK signal and a noise-like signal. The S/I parameter is the ratio of the desired signal 

power to the ASK interfering signal power, and the x-axis variable S/N is the ratio of the desired signal 

power to the total noise-like power (including the receiver noise and the noise-like interfering signal). 

The curves were generated by time-domain simulation. The IF bandwidth was 2.7 kHz. The audio 

(baseband) bandwidth was from 300 Hz to 3 kHz. The carrier and lower sideband were suppressed. 

The center frequency of the IF filter was assumed to be 1650 Hz above the suppressed carrier frequency. 

In the last part of the simulation, the signals were transformed to the frequency domain for calculating 

the AI, as described in Section 5.2.1. 

Figure 5-3 shows AI curves for a SSB voice receiver with ASK interference. There are six curves 

displayed. Each curve is a plot of AI vs. S/N. The top curve (labeled “No interference”) applies to the 

case in which there is no ASK interference. The other five curves are for cases with ASK interference. 

Each of those five curves is labeled with the S/I for that curve. As expected, each curve shows that the 

intelligibility (as indicated by the AI) generally improves as the S/N increases. As also expected, for a 

given S/N the intelligibility generally improves as the S/I increases. 

In the simulation, the ASK interfering signal had a bit rate of 800 bps, but other simulations showed the 

results to be insensitive to the bit rate. (However, at very high bit rates the interfering signal is noiselike, as specified in Section 3.3.3.) The ASK signal was tuned 500 Hz away from the suppressed carrier, 

but other simulations showed the results to be insensitive to variations of off-tuning within the same

frequency channel. 

Figure 5-4 shows AI curves for a SSB voice receiver with FSK interference. In the simulation, the FSK

interfering signal had a bit rate of 50 bps and an off-tuning of 500 Hz, but other simulations showed the 

results to be insensitive to the bit rate and to variations of off-tuning within the same frequency channel. 

The frequency deviation of the FSK signal was ± 200 Hz. 

5-10 
 JSC-CR-06-072 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-20 -10 0 10 20 30 40

S/N (dB)

AI

No interference S/I = 15 dB

S/I = 10 dB

S/I = 5 dB

S/I = 0 dB

S/I = -5 dB

S/I = -10 dB

Figure 5-3. AI vs. (S/N)in Curves for SSB Voice Receiver with ASK Interference 

5-11 
JSC-CR-06-072 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-20 -10 0 10 20 30 40

S/N (dB)

AI

No interference

S/I = 15 dB

S/I = 10 dB

S/I = 5 dB

S/I = 0 dB

S/I = -5 dB

S/I = -10 dB

Figure 5-4. AI vs. (S/N)in Curves for SSB Voice Receiver with FSK Interference

5-12 
 JSC-CR-06-072 

5.5 FM VOICE 

5.5.1 Description 

An FM signal has an RF carrier whose instantaneous frequency depends on the amplitude of the 

modulating waveform. More precisely, the deviation of the RF carrier frequency from its unmodulated 

state fc is made to be proportional to the amplitude of the modulating waveform m(t). For FM voice, 

m(t) is the voice waveform. However, properties of FM are more easily understood if the modulation is 

assumed to be sinusoidal. When the modulating waveform is a sinusoidal tone with frequency fm, the 

FM signal may be written: 

v(t) A cos[2 f t sin(2π f t)] = π c + β m

(5-6) 

where v(t) = waveform magnitude at time t, in V 

 t = time, in s 

 A = constant signal amplitude, in V

 fc = carrier frequency, in Hz 

β = modulation index (a unitless constant) 

 fm = frequency of modulation tone, in Hz 

The instantaneous frequency is the time derivative of the argument of the cosine function: 

2π f 2π f 2π f cos(2π f t) = c + β m m

(5-7) 

where f = instantaneous frequency, in Hz 

The maximum frequency deviation, which occurs when the modulating signal amplitude is at its 

maximum, is given by: 

( ) c m ∆f = f − f = β f max

(5-8) 

where ∆f = maximum (peak) frequency deviation, in Hz 

The bandwidth of the FM signal is found to be: 

5-13 
JSC-CR-06-072 

2( 1) 2( ) m m B = β + f = ∆f + f

(5-9) 

This equation states that the FM signal bandwidth is twice the sum of the maximum frequency deviation 

and the modulating frequency. This equation is called Carson’s Rule. For FM voice, the modulation 

frequency fm is the maximum baseband (audio) frequency. 

5.5.2 Performance 

5.5.2.1 FM Threshold 

The FM threshold is a point on the curve of output S/N vs. input S/N at which the curve changes 

dramatically. Above threshold, the input vs. output S/N relationship is linear. Below threshold, the 

output S/N falls off faster than the input S/N.

The threshold value is arbitrarily defined to be the value of input S/N for which the output S/N is 1 dB 

lower than the linear value. When the input S/N falls below this point, the system should be considered 

unusable. The threshold increases with an increase in the modulation index. Even though a large 

modulation index results in improved performance above threshold, this improvement comes with the 

price of a relatively higher threshold. 

Clearly, it is desirable for an FM receiver to have as low a threshold as possible. However, lowering the 

threshold implies a reduction in modulation index, which in turn deprives the system of the improved 

output S/N that a larger modulation index provides. System designers exploit these dependencies to 

tailor FM systems for particular applications. A system with a low modulation index – β equal to 0.5 or 

0.6, for example – will have a low threshold, enabling it to be used in conditions of marginal reception. 

This property may be important for tactical communications, where the low threshold is more important

than high fidelity. On the other hand, a system with a larger modulation index will be capable of higher 

fidelity, but at the expense of a higher threshold. This trade-off may make sense for commercial FM

systems that compete for listeners based, in part, on high-fidelity programs. 

The threshold for a particular system depends on more than the modulation index. There are several 

techniques available to designers for implementing FM demodulators. These implementations have 

some bearing on the point at which threshold occurs. The most reliable way to determine the threshold 

of an FM demodulator is by measurement. Along with the sensitivity, the threshold of an FM receiver 

should be part of the system specifications. 

5-14 
 JSC-CR-06-072 

FM systems that are designed for low threshold operation are often called narrowband systems. Since 

these systems have relatively low modulation indices, they also require relatively small operating 

bandwidths. Systems designed for high-fidelity operation are often called wideband systems. These 

systems have relatively large modulation indices, and require correspondingly large operating 

bandwidths. 

5.5.2.2 Narrowband Interference and Narrowband FM 

Narrowband interference may simply be due to signals from narrowband transmitters in the receiver 

environment. The self-interference due to multipath propagation in built-up areas – commonly referred 

to as fading – also tends to be narrowband in nature. 

Narrowband FM addresses these interference types in two ways. By maintaining a low threshold,

narrowband FM offers more operating margin than wideband FM. Moreover, since narrowband FM 

means less channel bandwidth, there are likely to be more available channels from which a nondegraded channel can be selected. 

5.5.2.3 Pre-emphasis and De-emphasis 

An FM system makes the best use of its operating bandwidth when the spectrum of the modulating 

signal is flat. An FM system can make it flat by applying a filter with the appropriate transfer function. 

This process is referred to as pre-emphasis. The demodulator cancels the effect of pre-emphasis by 

applying a filter with the complementary transfer function. This process is called de-emphasis. The 

success of this technique depends on a prior knowledge of the modulating signal spectrum.

Pre-emphasis and de-emphasis can improve the output S/N by 5 or 6 dB in some cases. This 

improvement should be interpreted as bringing a system closer to its optimum operating specification. If 

an FM system uses this technique, it is very likely that the sensitivity specification already includes the 

adjustment. 

5.5.3 Transfer Function 

For an FM demodulator operating above threshold with noise (or noise-like interference), the output S/N 

is related to the input S/N as follows: 





 



  +  



  =  



 2

out in 2

3 10 log β NS

N

S

(5-10) 

5-15 
JSC-CR-06-072 

where (S/N)out = output signal-to-noise power ratio, in dB 

(S/N)in = input signal-to-noise power ratio, in dB 

β = modulation index of the demodulator, unitless 

This expression demonstrates a fundamental property of FM – the ability to improve the output S/N by 

increasing the modulation index. Since an increase in modulation index corresponds to an increase in 

maximum frequency deviation, this improvement necessitates an increase in bandwidth. 

For a narrowband FM demodulator operating above threshold with narrowband FM interference, 

simulations show the transfer function to be: 

5

out in

 + 

 



  =  





N

S

N

S

(5-11) 

5.5.4 AI Curves 

Figures 5-5 and 5-6 show AI curves for an FM voice receiver. In these graphs, the term “interference” 

and the variable I refer to non-noise-like interference. Any noise-like interference power is simply 

assumed to be added (in mW) to the noise power. For example, suppose there are two interfering 

signals: an ASK signal and a noise-like signal. The S/I parameter is the ratio of the desired signal 

power to the ASK interfering signal power, and the x-axis variable S/N is the ratio of the desired signal 

power to the total noise-like power (including the receiver noise and the noise-like interfering signal). 

The curves were generated by time-domain simulation. The IF bandwidth was 16 kHz. The audio 

(baseband) bandwidth was from 300 Hz to 3.5 kHz. The peak frequency deviation was 5 kHz. These 

parameters imply a modulation index of β = 5/3.5 = 1.4. Note that the IF bandwidth is slightly smaller 

than the 2 (5 + 3.5) = 17 kHz specified by Carson’s rule. Pre-emphasis and de-emphasis were not 

included in the receiver model. In the last part of the simulation, the signals were transformed to the 

frequency domain for calculating the AI, as described in Section 5.2.1. 

Figure 5-5 shows AI curves for an FM voice receiver with ASK interference. In the simulation, the 

ASK interfering signal had a bit rate of 100 bps and was on-tune with the receiver. There are six curves 

displayed. Each curve is a plot of AI vs. S/N. The top curve (labeled “No interference”) applies to the 

case in which there is no ASK interference. The other five curves are for cases with ASK interference. 

Each of those five curves is labeled with the S/I for that curve. As expected, each curve shows that the 

intelligibility (as indicated by the AI) improves as the S/N increases. As also expected, for a given S/N

the intelligibility improves as the S/I increases. 

5-16 
 JSC-CR-06-072 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-10 0 10 20 30

S/N (dB)

AI

No interference S/I = 10 dB

S/I = 0 dB

S/I = -10 dB

S/I = -20 dB

S/I = -40 dB

Figure 5-5. AI vs. (S/N)in Curves for FM Voice Receiver with ASK Interference

5-17 
JSC-CR-06-072 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-10 0 10 20 30

S/N (dB)

AI

No interference S/I = 15 dB

S/I = 10 dB

S/I = 7 dB

S/I = 5 dB

S/I = 2 dB

Figure 5-6. AI vs. (S/N)in Curves for FM Voice Receiver with FSK Interference 

5-18 
 JSC-CR-06-072 

Figure 5-6 shows AI curves for an FM voice receiver with FSK interference. In the simulation, the FSK 

interfering signal had a bit rate of 50 bps and an off-tuning of 500 Hz, but other simulations showed the 

results to be insensitive to the bit rate and to variations of off-tuning within the same frequency channel. 

The frequency deviation of the FSK signal was ± 200 Hz. 

5.6 BROADCAST TELEVISION 

5.6.1 Description 

Vestigial sideband AM (with the carrier, one complete sideband, and a small portion of the other 

sideband) is used exclusively for broadcast television. Since the output is a combination of sound and 

picture, a single S/N value is inadequate to measure the receiver performance. For television receivers, 

it is common to specify the required performance subjectively. By experimentation, the subjective 

measurements can be related to the input S/N. In the TASO scoring procedure, observers are asked to 

rate picture quality on a scale of 1 to 6 as shown in Table 5-1. 

Table 5-1. TASO Score Definition

TASO 

Score Name Description

1 Excellent The picture is of extremely high quality, as good as you could desire. 

2 Fine The picture is of high quality providing enjoyable viewing. Interference is

perceptible. 

3 Passable The picture is of acceptable quality. Interference is not objectionable. 

4 Marginal The picture is poor in quality and you wish you could improve it. Interference 

is somewhat objectionable.

5 Inferior The picture is very poor but you could watch it. Definitely objectionable 

interference is present. 

6 Unusable The picture is so bad that you could not watch it. 

5.6.2 TASO Curves 

Figure 5-7 shows TASO curves for an analog broadcast TV receiver. There are three curves, one for 

each type of interference: ASK, noise-like, and TV interference. Each curve is a plot of TASO score vs. 

input S/I. As expected, each curve shows that the picture quality (as indicated by the TASO score) 

improves as the S/I increases. 

5-19 
JSC-CR-06-072 

1

2

3

4

5

6

0 10 20 30 40 50 6

S/I (dB)

TA

S

O

0

ASK interference

Noise-like interference

TV interference

 

Figure 5-7. TASO Score vs. (S/I)in Curves for Broadcast TV Receiver 

5-20 
 JSC-CR-06-072 

The curves were generated by collecting responses from a viewer panel.5-1 The receiver was a standard 

US commercial TV with an IF bandwidth of 5 MHz. The ASK interfering signal had a bit rate of 

100 bps and was on-tune with the receiver. The interfering TV signal was also on-tune with the 

receiver. 

 

5-1 H. Fine, A Further Analysis of TASO Panel 6 Data on Signal-to-Interference Ratios and Their Application to

Description of Television Service, T.R.R. Report No. 5.1.2, Washington, DC: FCC, 1 April 1960. 

5-21/5-22 
 JSC-CR-06-072 

SECTION 6 – DIGITAL DEMODULATOR 

This section describes the demodulator model for digital receivers. Performance measures for digital 

demodulators are discussed. The ratio of bit energy to noise power density (Eb/No) is defined. Transfer 

functions that relate output error rate to input Eb/No for noise-like interference are presented. Curves 

that specify output BER as a function of Eb/No and S/I are also displayed.

6.1 INTRODUCTION 

The purpose of the digital demodulator is to reconstruct the digital data that was used to modulate the 

carrier. In some cases, this digital data is a digitized audio or video waveform. The three carrier 

properties most commonly modified in the modulation process are phase, frequency, and amplitude. 

There are several properties of digital RF systems that affect system performance: 

Power Spectral Density and Bandwidth. The bandwidth required for a digital communication system

depends primarily on the data rate and the modulation type. It may be assumed that the data modulating 

the carrier is random. 

Since the data is random, the modulated signal constitutes a stochastic process. This process can be

characterized in the time domain by the autocorrelation function, which is calculated from the statistics 

of the modulated signal. The Fourier transform of the autocorrelation function is called the power 

spectral density (PSD) of the modulated signal. The PSD at a given frequency is the power per unit 

bandwidth (e.g., in W/Hz) at that frequency. 

Measurements and calculations of the PSD for digital signals show that the signal bandwidth is 

nominally equal to the bit rate or symbol rate. However, the precise bandwidth and spectral shape 

depends on the type of modulation. Some modulation types are quite spectrally efficient. The 

indiscriminate use of bandpass filtering to reduce the signal bandwidth is not practical, because a narrow 

filter bandwidth will distort the signal. 

Susceptibility to Noise and Interference. The susceptibility depends on the modulation type. A 

coherent demodulator generates a local waveform that is in phase with the received signal. Although it 

is more difficult and costly to implement than the noncoherent demodulator, it decreases susceptibility 

to noise and interference. Other modulation characteristics, such as the carrier property modulated (e.g., 

phase), also affect the susceptibility. As expected, performance is best for any modulation type when 

the desired signal power is significantly higher than the noise and interference power. Unlike analog 

6-1 
JSC-CR-06-072 

demodulators, digital demodulators exhibit a transition from unacceptable to virtually error-free 

performance over a relatively small range of input values.

Synchronization and Threshold. Digital demodulators require some level of synchronization to 

operate properly. At the very least, the demodulator must be able to identify the data bit boundaries. 

Additional levels of synchronization may also be required, since digital data is often organized into 

symbols, frames, packets, or other multi-bit structures. 

Synchronization algorithms in the receiver are designed to be particularly robust. Therefore, the 

required S/N for message transmission (in steady interference conditions) is usually also adequate for 

maintaining synchronization. If synchronization is lost for some reason – perhaps due to an interference 

pulse – it may be necessary to estimate the time needed to regain synchronous operation. If this time is 

not provided as part of the system specification, it is best determined by measurement or simulation. 

Soft Decision Demodulation. For a given received digital symbol waveform, a demodulator may make 

a definite decision as to which symbol is associated with that waveform. The output of this harddecision demodulator is a sequence of symbol values. In contrast, the output of a soft-decision

demodulator is a sequence of likelihood values. Each soft-decision value indicates the likelihood that a 

given symbol was sent, and a subsequent error correction device uses that likelihood to make a decision. 

Section 6 considers only hard-decision demodulators. For soft-decision systems, the demodulator and 

FEC decoder are treated as one combined module in Section 7. 

6.2 BINARY SYSTEMS 

In binary digital communications systems, the data can be represented as an information sequence 

consisting of binary digits (bits). Prior to modulation, additional bits may be added for error correction, 

synchronization, etc. Each bit has one of two possible values (0 or 1) and is associated with a distinct 

waveform. Consequently, bits have several properties that derive from their waveform representation: 

•

•

•

The bit duration Tb is the duration of the waveform associated with each bit. 

The bit rate (or data rate) Rb is the number of bits transmitted per second. 

The bit energy Eb at a given point in the system is the energy in the bit waveform at that point. 

The bit rate and bit duration are related by: 

b

b T

R 1 =

(6-1) 

6-2 
 JSC-CR-06-072 

where Rb = bit rate, in bits/s or Hz

 Tb = bit duration, in s 

The bit energy is related to the signal power S by: 

Eb = STb

(6-2) 

where Eb = bit energy, in J 

 S = signal power, in W

Noise within the receiver is assumed to have, at its point of origin, a constant power density at all 

frequencies. At the demodulator input, the total noise power is given by: 

N = NoB

(6-3) 

where N = noise power at the demodulator input, in W

No = one-sided noise power density at the demodulator input, in W/Hz 

 B = narrowest filter bandwidth preceding the demodulator, in Hz 

Based on the previous definitions, the ratio of bit energy to noise power density is given by: 

b

b b

R

B

N

S

N B

ST

N

E = = / o

(6-4) 

Just prior to making a decision on a given bit, the demodulator integrates all the power in the bit 

waveform. This action equivalently filters the signal with a bandwidth Rb, which is the signal 

bandwidth. If B exceeds Rb, then this operation improves the S/N by the factor B/Rb. Thus Eb/No can be 

viewed as an improved S/N (at the input to the decision circuit). In many cases, B is equal to the bit rate 

and Eb/No is equal to the S/N at the demodulator input. 

For each bit the demodulator must decide, based on the received signal, which of the two possible 

waveforms was transmitted. The waveforms are designed to be different enough that the correct 

decision will be made most of the time, even though the received signal has been degraded by the 

propagation environment. The standard measure of performance for the demodulator is the probability 

6-3 
JSC-CR-06-072 

of bit error, commonly referred to as the BER. Analytical expressions for the BER as a function of 

Eb/No are available for many digital demodulators, and are included in this section. 

6.3 M-ARY SYSTEMS 

In some digital communications systems, the bits are converted into multi-bit symbols prior to 

modulation. Each symbol has one of M possible values (0, 1, 2, ..., M-1) and is associated with a 

distinct waveform. Consequently, symbols have several properties that derive from their waveform

representation: 

•

•

•

The symbol duration Ts is the duration of the waveform associated with each symbol. 

The symbol rate R s is the number of symbols transmitted per second. 

The symbol energy Es at a given point in the system is the energy in the symbol waveform at that 

point. 

The number of possible symbols is given by: 

k M = 2

(6-5) 

where M = number of possible symbols 

 k = number of bits per symbol 

A system in which k = 1 and M = 2 is referred to as binary (Section 6.2). If k > 1, the system is referred 

to as M-ary. For example, an 8-ary system has three bits per symbol and M = 23

 = 8. The possible 

symbol values are 0, 1, 2, 3, 4, 5, 6, 7 and each of those values is associated with a distinct waveform. 

Equations 6-1, 6-2, and 6-4 can be generalized to the M-ary case: 

s

s T

R 1 =

(6-6) 

Es = STs

(6-7) 

6-4 
 JSC-CR-06-072 

s

s s

R

B

N

S

N B

ST

N

E = = / o

(6-8) 

Just prior to making a decision on a given symbol, the demodulator integrates all the power in the 

symbol waveform. This action equivalently filters the signal with a bandwidth Rs, which is nominally 

equal to the bandwidth of the desired signal. If B exceeds Rs, then this operation improves the S/N by 

the factor B/Rs. Thus, as in the binary case, Es/No can be viewed as an improved S/N (at the input to the 

decision circuit). In many cases, B is equal to the symbol rate and Es/No is equal to the S/N at the 

demodulator input. 

For each symbol the demodulator must decide, based on the received signal, which of the M possible 

waveforms was transmitted. The standard measure of performance for the demodulator is the 

probability of symbol error, commonly referred to as the symbol error rate (SER). Analytical 

expressions for the SER as a function of Es/No are available for many digital demodulators, and are 

included in this section. 

6.4 M-ARY SYSTEM PERFORMANCE IN BINARY TERMS 

It is often convenient to express the performance of an M-ary system in binary terms; i.e., in terms of

BER and Eb/No instead of SER and Es/No. This facilitates the performance evaluation of a demodulator 

by always expressing the input and output parameters in terms of the fundamental data unit – the bit. 

Because there are k bits per symbol, the duration of a symbol (Ts) is k times the duration of a bit (Tb). 

Equations 6-2 and 6-7 can then be combined to give: 

o o o No

E k

N

SkT

N

ST

N

Es s b b = = =

(6-9) 

The Eb/No and Es/No ratios are usually given in dB, in which case Equation 6-9 takes the form: 

o o dB

10 log( )



 



 = + k

N

E

N

Es b

(6-10) 

6-5 
JSC-CR-06-072 

The relationship between BER and SER is not so simple. Because there are k bits per symbol, a 

particular symbol error may result in only one bit error or as many as k bit errors (depending on which 

incorrect symbol was chosen). There are two common situations that occur. 

In the first situation, the M waveforms are orthogonal in signal space. An example is 8-ary FSK, for 

which 8 waveforms can be treated as 8 orthogonal vectors in an 8-dimensional vector space. Each

dimension is an independent frequency channel. This means that a random noise burst can occur with 

equal likelihood in any channel. Therefore, all incorrect symbols are equally likely to be chosen. In this 

case, the average number of bit errors (averaged over all possible incorrect symbol choices) can be 

calculated. The relationship between BER and SER is given by: 

SER

M

M BER 



 





− = 1

/ 2

(6-11) 

For example, the BER for an 8-ary FSK system would be 4/7 of the SER. 

In the second situation, the M waveforms are not orthogonal in signal space. An example is 8-ary PSK, 

for which 8 waveforms exist in a 2-dimensional vector space. In this situation, incorrect symbols

adjacent to the correct symbol are more likely to be chosen. In many such cases, the assigning of 

waveforms to symbols can be done so that adjacent symbols differ by only one bit. In these cases, a 

symbol error corresponds to one bit error and k-1 correct bits, so that 

k

SER BER =

(6-12) 

For example, the BER for an 8-ary PSK system would be 1/3 of the SER. 

If the noise and interference levels are high, then non-adjacent symbols may be chosen. Figure 6-1 

shows an analytical performance curve for 8-ary PSK, obtained by calculating the SER with an analytic 

equation and then using Equation 6-12. The corresponding simulation curve is also shown. Note that 

when the BER exceeds 0.2, Equation 6-12 is inaccurate (because of non-adjacent symbol choices). 

Equations 6-11 and 6-12 constitute upper and lower bounds, so that in general 

SER

M

M BER

k

SER 



 





− ≤ ≤

1

/ 2

(6-13) 

6-6 
 JSC-CR-06-072 

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

-30 -22 -14 -6 2 10

Eb/No (dB)

BE

R

Analytical

Simulation

100

10-1

10-2

10-3

10-4

10-5

Figure 6-1. Analytical and Simulation BER Curves for PSK Receiver with M = 8 

6.5 TYPES OF INTERFERENCE 

Section 3.3.3 presents a case in which interference becomes noise-like when it passes through the RF/IF 

section. In that case, the interfering signal bit rate or chip rate exceeds that of the receiver, and the 

interfering signal bandwidth exceeds the receiver bandwidth. Thus broadband interference may become

noise-like interference. To analyze its effect on performance, the noise-like interference power can 

simply be added (in W) to the other noise power within the receiver. 

At the other extreme, a signal that theoretically has zero bandwidth is the single-frequency sinusoid, or 

CW signal. Narrowband interference may often be modeled as CW interference, because the 

performance is not sensitive to the specific details of the interference modulation. Because the CW 

signal is a simple, unmodulated signal, analytical expressions for BER for the case of noise and CW

interference are available for some digital demodulators. In other cases for which a simulation must be 

performed, the use of a CW interfering signal simplifies the simulation. 

For all performance curves presented in this section, only two types of interference are considered – 

noise-like and CW. This assumption permits the curves in the handbook to be used in many practical 

6-7 
JSC-CR-06-072 

cases with broadband and narrowband interference. If a continuous interfering signal is neither 

broadband nor narrowband, the effect on the BER will generally fall between that of CW interference 

and that of noise. 

If an interfering signal is intermittent, the SER or BER will vary, and will therefore be less useful as a 

simple measure of performance. In this case, an analysis approach that considers the temporal 

properties of the interference should be devised (Section 3.4). 

6.6 PSK 

The PSK characterized here assumes coherent reception. In coherent reception, the receiver regenerates 

the carrier from the received signal and then uses the regenerated carrier as a reference for determining 

the phase of the modulating signal. Noncoherent PSK is presented in Section 6.7. 

6.6.1 Description 

6.6.1.1 Binary PSK 

Binary PSK (BPSK) is PSK with one bit per symbol (k = 1 and M = 2). The BPSK modulation 

represents logical 1 and logical 0 with two waveforms of the same frequency that are 180° out of phase. 

The voltage waveforms for 1 and 0 are: 

( ) cos(2 ) 1 v t A f t = π c

( ) cos(2 ) cos(2 ) 0 v t A f t A f t = π c + π = − π c

(6-15) 

where A = amplitude, in V 

 fc = carrier frequency, in Hz 

 t = time, in s 

Alternatively, the waveforms may be expressed as the product of the modulating (baseband) signal and 

the RF carrier: 

v(t) b(t) A cos(2 f t) = π c

(6-16) 

where b(t) = 1 for logical 1 and b(t) = -1 for logical 0. 

6-8 
 JSC-CR-06-072 

For noise and noise-like interference, the BER is given by: 















 =

o

erfc

2

1

N

E

BER b

(6-17) 

where erfc(x) is the complementary error function. 

6.6.1.2 Quadrature PSK 

Quadrature PSK (QPSK) is PSK with two bits per symbol (k = 2 and M = 4). One of four possible 

waveforms is transmitted: 





 



 = +

2 ( ) cos 2 π

v t A π f t i c

(6-18) 

where i = 0, 1, 2, or 3. The waveforms differ only in their phase. The phase angles are specified to have 

a uniform spacing of π/2 radians. Since there are four possible waveforms, it is possible to represent 

two bits of data per symbol. 

The QPSK signal is essentially two binary PSK signals in phase quadrature. QPSK is a special M > 2 

case in which an exact expression for the BER can be given. For noise and noise-like interference, the 

BER for QPSK is the same as for BPSK: 















 =

o

erfc

2

1

N

E

BER b

(6-19) 

The SER is given by (Reference 4-2): 































 −































 =

o o

erfc

4

1 erfc 1

N

E

N

E SER b b

(6-20) 

6-9 
JSC-CR-06-072 

6.6.1.3 M-ary PSK for M > 4 

This is an extension of 2-phase and 4-phase modulation to the general case of M-phase modulation with 

k bits per symbol. One of M possible waveforms is transmitted: 





 



 = +

M

v t A f t i c

π

π

2 ( ) cos 2

(6-21) 

where i = 0, 1, 2, ..., M – 1. The waveforms differ only in their phase. The phase angles are specified to 

have a uniform spacing of 2π/M radians. 

For noise and noise-like interference, the SER for M > 4 is approximated by (Reference 4-2): 















 



 



 = N M

E SER k b π erfc sin

o

(6-22) 

Equation 6-12 can be used to determine the corresponding BER. 

6.6.2 BER Curves 

Figures 6-2, 6-3, 6-4, and 6-5 show BER curves for a PSK receiver with M =2, M = 4, M = 8, and 

M = 16, respectively. In these graphs, the term “interference” and the variable I refer to CW

interference. Any noise-like interference power is simply assumed to be added (in mW) to the noise 

power. For example, suppose there are two interfering signals: a CW signal and a noise-like signal. 

The S/I parameter is the ratio of the desired signal power to the CW interfering signal power, and the 

x-axis variable Eb/No is calculated with Equation 6-4, where N is the total noise-like power (including 

the receiver noise and the noise-like interfering signal). 

The curves were generated from analytic expressions, including Equations 6-12, 6-17, 6-19, and 6-22. 

Analytic expressions for cases involving CW interference required numerical integration. 

Each figure displays six curves. Each curve is a plot of BER vs. Eb/No. The curve labeled “No 

interference” applies to the case in which there is no CW interference. The other five curves are for 

cases with CW interference. Each of those five curves is labeled with the S/I for that curve. As

expected, each curve shows that the BER decreases as the Eb/No increases. As also expected, for a given 

Eb/No the BER decreases as the S/I increases. 

6-10 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-4 0 4 8 12 16 20 24

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 6 dB

S/ I = 4 dB

S/ I = 3 dB

S/ I = 2 dB

No interference

Figure 6-2. BER vs. Eb/No Curves for PSK Receiver (M = 2) with CW Interference 

6-11 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-4 0 4 8 12 16 20 24

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 5 dB

S/ I = 15 dB S/ I = 6 dB

S/ I = 8 dB

No interference

Figure 6-3. BER vs. Eb/No Curves for PSK Receiver (M = 4) with CW Interference 

6-12 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

0 4 8 12 16 20 24 28

Eb/No (dB)

BE

R

S/ I = 18 dB

S/ I = 14 dB

S/ I = 12 dB

S/ I = 11 dB

S/ I = 10 dB

10-1

10-2

10-3

10-4

10-5

10-6

No interference

Figure 6-4. BER vs. Eb/No Curves for PSK Receiver (M = 8) with CW Interference 

6-13 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26 30 34

Eb/No (dB)

BER

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 24 dB

S/ I = 20 dB

S/ I = 18 dB

S/ I = 17 dB

S/ I = 16 dB

No interference

Figure 6-5. BER vs. Eb/No Curves for PSK Receiver (M = 16) with CW Interference 

6-14 
 JSC-CR-06-072 

6.7 DPSK 

DPSK is a noncoherent modulation scheme. In coherent reception, the receiver regenerates the carrier 

from the received signal and then uses the regenerated carrier as a reference for determining the phase of 

the modulating signal. In DPSK, the receiver does not regenerate the carrier. The demodulator 

compares the phase of the current bit waveform to the phase of the previous bit waveform rather than to 

the carrier. Although DPSK permits a simpler receiver implementation, the error performance is not as 

good as the coherent case. Coherent PSK is presented in Section 6.6. 

6.7.1 Description 

The waveforms for DPSK are the same as for PSK (Equations 6-15, 6-18, and 6-21). However, the 

phase term in the equation has a different interpretation. In coherent PSK, a waveform has the phase 

i2π/M when the corresponding symbol value is i. In DPSK, two succeeding waveforms have the phase 

difference i2π/M when the corresponding symbol value is i. 

For noise and noise-like interference, the BER for binary DPSK is given by: 















 = −

o

exp 2

1

N

E

BER b

(6-23) 

For DPSK with M > 2, the SER for noise and noise-like interference is given by (Reference 4-2): 















 



 



 = N M

E SER k b

2

erfc sin

o

π

(6-24) 

Equation 6-12 can be used with Equation 6-24 to estimate the BER for M-ary DPSK. 

6.7.2 BER Curves 

Figures 6-6, 6-7, and 6-8 show BER curves for a DPSK receiver with M =2, M = 4, and M = 8, 

respectively. In these graphs, the term “interference” and the variable I refer to CW interference. Any 

noise-like interference power is simply assumed to be added (in mW) to the noise power. For example, 

suppose there are two interfering signals: a CW signal and a noise-like signal. The S/I parameter is the 

ratio of the desired signal power to the CW interfering signal power, and the x-axis variable Eb/No is 

6-15 
JSC-CR-06-072 

calculated with Equation 6-4, where N is the total noise-like power (including the receiver noise and the 

noise-like interfering signal). 

The curves were generated by simulation. Each figure displays six curves. Each curve is a plot of BER 

vs. Eb/No. The curve labeled “No interference” applies to the case in which there is no CW interference. 

The other five curves are for cases with CW interference. Each of those five curves is labeled with the 

S/I for that curve. As expected, each curve shows that the BER decreases as the Eb/No increases. As 

also expected, for a given Eb/No the BER decreases as the S/I increases. 

6.8 FSK 

6.8.1 Description 

FSK uses M different frequencies to transmit the digital data. These signaling frequencies are equally 

spaced and centered about the RF carrier frequency. The waveform of the FSK signal may be 

expressed: 

v (t) A cos(2 f t) i = π i

(6-25) 

where fi = symbol frequency, in Hz. 

The separation between adjacent frequencies (∆f = fi – fi-1) affects the required bandwidth and the BER. 

The optimum (smallest) BER occurs when the signaling frequencies are orthogonal. The frequencies 

are orthogonal when 2∆f is an integer multiple of the symbol rate R s. 

Coherent reception means that the demodulator is phase-locked to the received signal. For noise and 

noise-like interference, the BER for coherent FSK (CFSK) is (Reference 4-2): 















−  = 2 o

erfc

2

1

N

M k E

BER b

(6-26) 

This equation assumes orthogonal signaling frequencies.

6-16 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26 30

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 1 dB

S/ I = 9 dB

S/ I = 5 dB

S/ I = 3 dB

S/ I = 2 dB

No interference

 

Figure 6-6. BER vs. Eb/No Curves for DPSK Receiver (M = 2) with CW Interference 

6-17 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

0 4 8 12 16 20 24 28

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 11 dB S/ I = 9 dB

S/ I = 17 dB

No interference S/ I = 13 dB

Figure 6-7. BER vs. Eb/No Curves for DPSK Receiver (M = 4) with CW Interference 

6-18 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26 30

Eb/No (dB)

BE

R

S/ I = 20 dB

S/ I = 23 dB S/ I = 18 dB

S/ I = 16 dB

S/ I = 17 dB

10-1

10-2

10-3

10-4

10-5

10-6

No interference

Figure 6-8. BER vs. Eb/No Curves for DPSK Receiver (M = 8) with CW Interference 

6-19 
JSC-CR-06-072 

The FSK demodulator can be simplified by employing two bandpass filters to noncoherently detect the 

signaling frequencies. Such a scheme results in degraded performance relative to coherent detection. 

For noise and noise-like interference, the SER for noncoherent FSK (NCFSK) is (Reference 4-2): 

















+ − − − +

− = ∑ −

−

=

+

o

1

1

1

1 exp1

1

!( 1 )!

( 1)! ( 1) N

E

j

jk

j M j j

M SER b

M

j

j

(6-27) 

Equation 6-11 can be used with Equation 6-27 to estimate the BER for noncoherent FSK. For the binary 

case (k = 1 and M = 2), Equation 6-27 reduces to: 















 = −

o 2

1

exp 2

1

N

E

BER b

(6-28) 

6.8.2 BER Curves 

Figure 6-9 shows BER curves for a binary coherent FSK receiver. Figures 6-10, 6-11, and 6-12 show 

BER curves for a noncoherent FSK receiver with M =2, M = 4, and M = 8, respectively. In these graphs, 

the term “interference” and the variable I refer to CW interference. Any noise-like interference power is 

simply assumed to be added (in mW) to the noise power. For example, suppose there are two interfering 

signals: a CW signal and a noise-like signal. The S/I parameter is the ratio of the desired signal power 

to the CW interfering signal power, and the x-axis variable Eb/No is calculated with Equation 6-4, where 

N is the total noise-like power (including the receiver noise and the noise-like interfering signal). 

The curves were generated from analytic expressions. The expressions involving CW interference 

required numerical integration. Each figure displays six curves. Each curve is a plot of BER vs. Eb/No. 

The curve labeled “No interference” applies to the case in which there is no CW interference. The other 

five curves are for cases with CW interference. Each of those five curves is labeled with the S/I for that 

curve. In all cases, the CW interference was close in frequency to one of the M signaling frequencies. 

As expected, each curve shows that the BER decreases as the Eb/No increases. As also expected, for a 

given Eb/No the BER decreases as the S/I increases. 

6-20 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 5 dB

S/ I = 20 dB

No interference

Figure 6-9. BER vs. Eb/No Curves for Coherent FSK Receiver (M = 2) with CW Interference 

6-21 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

4 8 12 16 20

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 15 dB S/ I = 5 dB

No interference

Figure 6-10. BER vs. Eb/No Curves for Noncoherent FSK Receiver (M = 2) with CW Interference 

6-22 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 5 dB

S/ I = 2 dB

S/ I = 10 dB

No interference

Figure 6-11. BER vs. Eb/No Curves for Noncoherent FSK Receiver (M = 4) with CW Interference 

6-23 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 5 dB

S/ I = 2 dB

S/ I = 10 dB

No interference

Figure 6-12. BER vs. Eb/No Curves for Noncoherent FSK Receiver (M = 8) with CW Interference 

6-24 
 JSC-CR-06-072 

6.9 MINIMUM-SHIFT KEYING 

6.9.1 Description 

The minimum-shift keying (MSK) waveform can be represented as an FSK waveform with the two 

signaling frequencies separated by Rb/2. This is the minimum separation that allows orthogonality. 

MSK can also be represented as a form of quadrature-modulated PSK (i.e., QPSK), with the additional 

specifications that 

•

•

•

The quadrature carriers are offset in time by Tb (offset QPSK or OQPSK)

Even bits appear on the in-phase channel and odd bits appear on the quadrature channel 

The baseband (modulating) pulses are sinusoidal, rather than rectangular. 

Because of the waveform complexity, it is necessary to give the waveform of the entire bit sequence 

rather than the waveform for a single bit. The entire MSK waveform is (Reference 4-2): 

( ) ( 2 ) cos(2 ) ( [2 1] ) sin(2 ) 2 2 1 v t A I u t nT f t A I u t n T f tc

n

c n b

n

n b π π 



 





+ − + 



 



 = ∑ − ∑

∞

=−∞

+

∞

=−∞

(6-29) 

where ∑ represents the sequence of even-numbered information bits (I

∞

n=−∞

n I 2 2n = ±1) 

∑

∞

=−∞

+

n

n I 2 1 represents the sequence of odd-numbered information bits (I2n+1 = ±1) 

The sinusoidal pulse-shaping function is: 

b Tb u(t) = sin(π t / 2T ) 0 ≤ t ≤ 2

(6-30) 

Note that u(t) extends over two bit intervals. These MSK properties eliminate phase discontinuities that 

would otherwise occur when the waveform changes state in accordance with the modulating signal. For 

this reason, MSK is more spectrally efficient than either standard QPSK or OQPSK with rectangular 

modulating pulses. 

This expression for the MSK waveform clearly shows the information bits, the pulse-shaping function, 

and the quadrature carriers. A variety of modulation types can be derived from OQPSK by varying the 

6-25 
JSC-CR-06-072 

pulse-shaping function. Some communications systems are capable of varying the pulse-shaping 

function in response to operational conditions.

The BER for MSK is the same as for standard QPSK: 















 =

o

erfc

2

1

N

E

BER b

(6-31) 

6.9.2 BER Curves 

Figure 6-13 shows BER curves for an MSK receiver. In this graph, the term “interference” and the 

variable I refer to CW interference. Any noise-like interference power is simply assumed to be added 

(in mW) to the noise power. For example, suppose there are two interfering signals: a CW signal and a 

noise-like signal. The S/I parameter is the ratio of the desired signal power to the CW interfering signal 

power, and the x-axis variable Eb/No is calculated with Equation 6-4, where N is the total noise-like

power (including the receiver noise and the noise-like interfering signal). 

The curves are the same as the QPSK curves, which were generated from analytic expressions. The 

expressions involving CW interference required numerical integration. The figure displays six curves. 

Each curve is a plot of BER vs. Eb/No. The curve labeled “No interference” applies to the case in which 

there is no CW interference. The other five curves are for cases with CW interference. Each of those 

five curves is labeled with the S/I for that curve. As expected, each curve shows that the BER decreases 

as the Eb/No increases. As also expected, for a given Eb/No the BER decreases as the S/I increases.

6.10 QUASI-BANDLIMITED MSK 

6.10.1 Description

Quasi-bandlimited MSK (QBL-MSK) is a variation of MSK. In the MSK waveform (Equations 6-29 

and 6-30), a pulse shaping function u(t) is applied to the normally rectangular pulses that represent the 

input bit sequence. The pulse-shaping function is used to provide smooth transitions between bit 

periods, as seen in the time domain. The result of this pulse-shaping, as seen in the frequency domain, is 

that more of the signal power is concentrated around the RF carrier. This in turn allows for greater 

bandwidth efficiency, which is the ratio of the bit rate to the bandwidth required for transmission at that 

bit rate. 

6-26 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-4 0 4 8 12 16 20 24

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 5 dB

S/ I = 15 dB S/ I = 6 dB

S/ I = 8 dB

No interference

Figure 6-13. BER Curves for an MSK Receiver with CW Interference 

6-27 
JSC-CR-06-072 

Bandwidth efficiency is indirectly related to error performance. While it may be useful to compare the 

error performance (BER vs. Eb/No) of modulation types assuming no band-limiting, any practical system

must operate within a predetermined bandwidth. This constraint favors modulation types that have high 

bandwidth efficiency. For a given bit rate, bandwidth efficiency can be adjusted by employing different 

pulse-shaping functions. 

The wide variety of pulse-shaping functions that have been studied can be classified as constantenvelope or varying-envelope, referring to the modulated RF signal. The type of envelope affects the 

power efficiency of the communications system. A signal with a non-constant envelope requires linear 

amplification. If the amplification is not linear, the bandwidth of the signal will be increased. Linear 

amplification is not required for signals with a constant-envelope. A typical amplifier is linear over only 

part of its operating range. Consequently, a signal with a non-constant envelope can utilize only a 

portion of the available operating range of an amplifier. 

Certain pulse-shaping functions that exhibit high bandwidth efficiency result in modulated signals that 

have non-constant envelopes. This property reduces the power efficiency of the transmitter. Either a 

more linear or a more powerful amplifier will be required, or else the transmitted power will be reduced. 

Bandwidth efficiency and power efficiency are competing interests for signal designers. Consideration 

of these interests has led to the development of modulation types that offer a compromise. QBL-MSK is 

an example of such a modulation type. It offers relatively good bandwidth efficiency, while maintaining 

the constant-envelope property. Thus, power efficiency is not compromised. 

The waveform of the MSK signal includes a sinusoidal pulse-shaping function u(t). For QBL-MSK, the 

pulse-shaping function is: 

b b

n

b

b T t T

t T

t T

u t 2 2

/

sin( / ) ( ) − ≤ ≤ 













 = π

π

(6-32) 

where typically n = 3 . 

The result of employing this pulse-shaping technique to bandlimit the signal is a modest (2 dB or less) 

improvement in error-performance with respect to standard MSK. The precise improvement depends on 

the precise bandwidth. 

6-28 
 JSC-CR-06-072 

6.10.2 BER Curves 

Figure 6-14 shows BER curves for a QBL-MSK receiver with n = 3 (Equation 6-32). In this graph, the 

term “interference” and the variable I refer to CW interference. Any noise-like interference power is 

simply assumed to be added (in mW) to the noise power. For example, suppose there are two interfering 

signals: a CW signal and a noise-like signal. The S/I parameter is the ratio of the desired signal power 

to the CW interfering signal power, and the x-axis variable Eb/No is calculated with Equation 6-4, where 

N is the total noise-like power (including the receiver noise and the noise-like interfering signal). 

The curves were generated by simulation. The figure displays three curves. Each curve is a plot of BER 

vs. Eb/No. The curve labeled “No interference” applies to the case in which there is no CW interference. 

The other two curves are for cases with CW interference. Each of those two curves is labeled with the 

S/I for that curve. As expected, each curve shows that the BER decreases as the Eb/No increases. As 

also expected, for a given Eb/No the BER decreases as the S/I increases. 

6.11 ASK 

6.11.1 Description

ASK uses M different equally spaced amplitudes to transmit the digital data. Each M-ary symbol 

represents k bits of information. The waveform of the ASK signal may be expressed: 

v (t) A u(t) cos(2 f t) i = i π

(6-33) 

where Ai = amplitude for symbol i, in V 

u(t) is a pulse-shaping function. If the pulses are rectangular, then u(t) = 1. Other pulse shapes may be 

employed to reduce the required bandwidth of the transmitted signal. 

Coherent reception means that the demodulator is phase-locked to the received signal. If the system is 

coherent, the detector can distinguish phase reversals. In that case, A and –A are two distinct 

amplitudes. A unipolar system uses only non-negative amplitudes, whereas a bipolar system uses 

positive and negative amplitudes. For example, a 4-ary unipolar system has amplitudes 0, a, 2a, 3a. A 

4-ary bipolar system with the same amplitude spacing (a) has amplitudes -3a/2, -a/2, a/2, 3a/2. 

6-29 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-4 0 4 8 12 16 20 24

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 6 dB

No interference

Figure 6-14. BER vs. Eb/No Curves for QBL-MSK Receiver with CW Interference 

6-30 
 JSC-CR-06-072 

For noise and noise-like interference, the SER for coherent unipolar M-ary ASK is (Reference 4-2): 

















− −

− =

o 2( 1)(2 1)

3 erfc 1

N

E

M M

k

M

M SER b

(6-34) 

For noise and noise-like interference, the SER for coherent bipolar M-ary ASK is (Reference 4-2): 

















−

− =

o

2 ( 1)

3 erfc 1

N

E

M

k

M

M SER b

(6-35) 

Equation 6-12 can be used with either Equation 6-34 or 6-35 to determine the corresponding BER. 

Coherent unipolar binary ASK (with amplitudes 0 and a) is also known as on-off keying (OOK). Note

that coherent bipolar binary ASK (with amplitudes –a/2 and a/2) is the same as binary PSK and 

Equation 6-35 reduces to Equation 6-17 in the binary case. 

ASK can also be detected noncoherently. In this case, the waveforms must be unipolar. The 

noncoherent demodulator implementation is simpler, but its performance is not as good as that of 

coherent detection. 

6.11.2 BER Curves 

Figures 6-15 and 6-16 show BER curves for a coherent unipolar ASK receiver with M = 2 and M = 4, 

respectively. Figures 6-17 and 6-18 show BER curves for a coherent bipolar ASK receiver with M = 4 

and M = 8, respectively. Figure 6-19 shows BER curves for a noncoherent ASK receiver with no CW

interference. In these graphs, the term “interference” and the variable I refer to CW interference. Any 

noise-like interference power is simply assumed to be added (in mW) to the noise power. For example, 

suppose there are two interfering signals: a CW signal and a noise-like signal. The S/I parameter is the 

ratio of the desired signal power to the CW interfering signal power, and the x-axis variable Eb/No is 

calculated with Equation 6-4, where N is the total noise-like power (including the receiver noise and the 

noise-like interfering signal). 

6-31 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 6 dB

S/ I = 4 dB

S/ I = 3 dB

S/ I = 2 dB

No interference

Figure 6-15. BER vs. Eb/No Curves for Coherent Unipolar ASK Receiver 

(M = 2) with CW Interference 

6-32 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

6 10 14 18 22 26 30

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 9 dB

S/ I = 6 dB

S/ I = 4 dB

S/ I = 3 dB

S/ I = 2 dB

No interference

Figure 6-16. BER vs. Eb/No Curves for Coherent Unipolar ASK Receiver 

(M = 4) with CW Interference 

6-33 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

0 4 8 12 16 20 24 28

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 6 dB

S/ I = 4 dB

S/ I = 3 dB

S/ I = 2 dB

No interference

Figure 6-17. BER vs. Eb/No Curves for Coherent Bipolar ASK Receiver 

(M = 4) with CW Interference 

6-34 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

4 8 12 16 20 24 28 32

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 6 dB

S/ I = 2 dB

S/ I = 10 dB S/ I = 3 dB

S/ I = 4 dB

No interference

Figure 6-18. BER vs. Eb/No Curves for Coherent Bipolar ASK Receiver 

(M = 8) with CW Interference 

6-35 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

6 10 14 18 22 2

Eb/No (dB)

BE

R

6

10-1

10-2

10-3

10-4

10-5

10-6

M = 2 M = 4 M = 8

Figure 6-19. BER vs. Eb/No Curves for Noncoherent ASK Receiver 

6-36 
 JSC-CR-06-072 

The curves were generated from analytic expressions. The expressions involving CW interference 

required numerical integration. Figures 6-15 through 6-18 display six curves of BER vs. Eb/No. The 

curve labeled “No interference” applies to the case in which there is no CW interference. The other five 

curves are for cases with CW interference. Each of those five curves is labeled with the S/I for that 

curve. As expected, each curve shows that the BER decreases as the Eb/No increases. As also expected, 

for a given Eb/No the BER decreases as the S/I increases. 

6.12 QUADRATURE AMPLITUDE MODULATION 

6.12.1 Description

The quadrature amplitude-modulation (QAM) technique employs two carrier signals in phase 

quadrature. The QAM-modulated waveform can be expressed: 

v (t) A u(t) cos(2 f t) B u(t)sin(2 f t) i = i π c + i π c

(6-36) 

where Ai, Bi = in-phase and quadrature amplitudes for symbol i, in V 

u(t) is a pulse-shaping function. If the pulses are rectangular, then u(t) = 1. Other pulse shapes may be 

employed to reduce the required bandwidth of the transmitted signal. Usually, there are 2m values of Ai

and 2m values of Bi, where m is an integer. Thus, the value of M might be (22

)( 22) = 16, (23)( 23) = 64, 

(24

)( 24) = 256, etc. Equation 6-5 ( k M = 2 ) is still applicable, so M = 256 when the number of bits per 

symbol (k) is 16. 4-ary QAM is the same as QPSK. 

For noise and noise-like interference, the SER for a QAM system is approximated by (Reference 4-2): 

















− =

o 2( 1)

3 2 erfc

N

E

M

k SER b

(6-37) 

QAM can be viewed as two coherent bipolar ASK signals impressed on the quadrature channels. 

Therefore, the SER with CW interference can be approximately determined by using Equation 6-35 

(with M = MQAM ) to determine the SER for each coherent bipolar ASK signal. Then the SER for the 

QAM signal is given by: 

6-37 
JSC-CR-06-072 

2 SER = 1−[1− SER(ASK)]

(6-38) 

where SER(ASK) = SER from Equation 6-35. 

Equation 6-12 can be used with either Equation 6-37 or 6-38 to determine the corresponding BER. 

6.12.2 BER Curves 

Figures 6-20 and 6-21 show BER curves for a QAM receiver with M = 16 and M = 64, respectively. In 

these graphs, the term “interference” and the variable I refer to CW interference. Any noise-like 

interference power is simply assumed to be added (in mW) to the noise power. For example, suppose 

there are two interfering signals: a CW signal and a noise-like signal. The S/I parameter is the ratio of

the desired signal power to the CW interfering signal power, and the x-axis variable Eb/No is calculated 

with Equation 6-4, where N is the total noise-like power (including the receiver noise and the noise-like 

interfering signal). 

The curves were generated from analytic expressions. Each figure displays four curves. Each curve is a 

plot of BER vs. Eb/No. The curve labeled “No interference” applies to the case in which there is no CW

interference. The other three curves are for cases with CW interference. Each of those three curves is 

labeled with the S/I for that curve. As expected, each curve shows that the BER decreases as the Eb/No

increases. As also expected, for a given Eb/No the BER decreases as the S/I increases.

6-38 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

No interference

S/ I = 20 dB

S/ I = 10 dB

S/ I = 5 dB

Figure 6-20. BER vs. Eb/No Curves for QAM Receiver (M = 16) with CW Interference 

6-39 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

6 10 14 18 22 26 30

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 5 dB

S/ I = 20 dB

No interference

Figure 6-21. BER vs. Eb/No Curves for QAM Receiver (M = 64) with CW Interference 

6-40 
 JSC-CR-06-072 

SECTION 7 – FEC DECODER 

This section describes the FEC decoder model for digital receivers. Decoder inputs and outputs are 

described for both hard-decision and soft-decision schemes. The properties of various block and 

convolutional decoders are specified. Some general procedures for calculating output BER are 

presented, including the case in which a receiver has concatenated FEC decoders. For hard-decision 

systems, curves that specify output BER as a function of input BER are displayed. For soft-decision 

systems, the demodulator and FEC decoder are treated as one combined module, and curves that specify 

output BER as a function of input Eb/No are displayed. 

7.1 INTRODUCTION 

The purpose of the FEC decoder is to correct bit errors that are present in the demodulated data. These 

errors are the result of noise and interference in the channel. The errors are corrected by applying the 

decoding algorithm associated with the encoding algorithm used by the transmitter. Ideally, the bit 

sequence at the FEC decoder output is identical to the original bit sequence at the FEC encoder input. 

FEC encoding involves adding redundancy, in the form of extra bits, to the information data. The FEC 

decoder removes this redundancy during the decoding process. There are two basic types of FEC code: 

block codes and convolutional codes. 

There are limits to the capabilities of error-correction codes. In general, the reduction in BER is directly 

related to the amount of redundancy added. This section provides transfer functions and analysis 

techniques that quantify the performance of FEC decoders. This section also introduces some of the 

terminology that is specific to FEC technology and analysis. 

7.2 HARD DECISION VS. SOFT DECISION 

The FEC decoder gets its input from the digital demodulator. A hard-decision demodulator makes a 

definite decision on each input bit waveform (which may be degraded with noise and interference) and 

then outputs that decision as a 0-bit or a 1-bit. This output bit waveform may use two voltage levels, say 

-V and +V, to represent a logical 0-bit or 1-bit. There is nothing in the output data that indicates how

reliable the demodulator decisions are. 

A soft-decision demodulator does not make a definite decision on a given input bit waveform. Instead, 

it outputs a bit likelihood indicator; i.e., a value that indicates the likelihood that a 0-bit or 1-bit was 

sent. For example, the bit likelihood indicator for a given bit may be an integer value between 0 and 7, 

7-1 
JSC-CR-06-072 

where 0 represents a very high likelihood that a 0-bit was sent and 7 represents a very low likelihood 

that a 0-bit was sent. The intermediate values from 1 to 6 represent intermediate likelihoods that a 0-bit 

was sent. For example, a value of 6 means that a 1-bit was probably sent but that there is some

unreliability in that decision. 

FEC decoders can be designed to work with either type of input. The decoder examines bit sequences 

for errors and, if it detects any, it makes correcting adjustments to the data. If the decoder input is a softdecision input, the decoder can use the reliability information to improve its correction algorithm. Softdecision demodulation and decoding can significantly improve the performance of a system that 

employs FEC coding, while adding very little to the system complexity and cost. 

7.3 DECODER PERFORMANCE 

For a hard-decision system, the input to the FEC decoder is a sequence of bits, including information 

bits and redundancy bits. Because the decoder removes the redundancy bits, the output bit rate is lower 

than the input bit rate. Typically, the decoder corrects some bit errors and the output BER is less than 

the input BER. The measure of performance in this handbook for such an FEC decoder is the output 

BER expressed as a function of the input BER. The output BER is sometimes referred to as the residual 

error rate. 

For a soft-decision system, the input to the FEC decoder is a sequence of bit likelihood indicators, one 

for each information bit and redundancy bit. As in hard decision, there is one decoder output bit for 

each information bit, and a bit error occurs whenever the decoder output bit is not equal to the 

corresponding original information bit. In this case, there is no BER at the decoder input because 

definite bit decisions have not been made at that point. For such a system, the most direct measure of 

performance combines demodulator and decoder effects. It specifies the decoder output BER as a 

function of Eb/No and S/I at the demodulator input. 

7.4 TEMPORAL FLUCTUATIONS 

If an undesired signal is intermittent, bit errors tend to be more numerous during the interference dwells, 

and less numerous between the dwells. If the interference is of sufficient strength and duration, the FEC 

decoder may be overloaded and the BER may approach 1⁄2 during a dwell, which means the decoder 

output is completely random. Over the long term, this condition is likely to manifest itself as error 

bursts, separated by periods of relatively error-free operation. 

However, the overload status of the decoder depends on how the interference dwell duration “matches 

up” with decoder properties such as code word duration. Interleaving (described in Section 7.5.8) can 

7-2 
 JSC-CR-06-072 

be used with hard-decision or soft-decision, block or convolutional codes to effectively convert long 

interference pulses into shorter, more uniformly distributed pulses. It should be noted that some FEC 

codes are designed specifically to handle burst errors. Such a code may be concatenated with a second 

code designed to combat random errors, in order to provide more complete error protection. 

In both block and convolutional decoding, output bit errors tend to appear in clusters. However, the 

output BER, which is a long-term average quantity, is still a valid and useful measure of performance. 

7.5 BLOCK CODES 

7.5.1 General Concepts and Terminology

A block code is a mapping between two sets of bits: the information bits and the coded bits. The 

information bits are grouped into blocks with k bits per block. These information blocks are sometimes 

referred to in technical literature as messages. There are 2k

 possible messages. The coded bits are 

grouped into blocks with n bits per block, where . These code blocks are usually called code 

words. There are 2

n > k

n

 possible code words. The set of code words is called the code set, or simply the 

code. 

The code mapping is the algorithm that associates every possible message to a unique code word. Thus, 

block encoding involves the addition of n - k bits of redundancy to each message. Such a code is 

referred to as an (n, k) code. The ratio defined as R = k / n is the code rate. The mapping is one-to-one, 

which means that there are only 2k

 valid code words, even though there are 2n possible code words. The 

decoder exploits the fact that only some of the possible n-bit code words are valid, through a process 

called maximum likelihood decoding. 

The weight of a code word is defined as the number of 1-bits in the code word. If all of the code words 

in the set have the same weight, the code is called an equal-weight code. 

The distance between two code words is the number of bit positions in which the two code words differ. 

The minimum distance of a code set, which is designated d, is the smallest distance between all code word 

pairs in the set. In general, a larger minimum distance implies a more powerful error-correction 

capability. 

A block code is linear if the modulo-2 sum of any two code words is also a code word. This property 

reduces the complexity of the FEC encoder. 

7-3 
JSC-CR-06-072 

A systematic code is one in which each codeword is a concatenation of the message it represents with 

r = n − k additional parity bits. This property reduces the complexity of the FEC encoder. 

An example of a (7, 4) linear systematic code is shown in Table 7-1. 

Table 7-1. Linear, Systematic (7, 4) Block Code 

Message Code Word

0000 0000000

0001 0001011

0010 0010110

0011 0011101

0100 0100111

0101 0101100

0110 0110001

0111 0111010

1000 1000101

1001 1001110

1010 1010011

1011 1011000

1100 1100010

1101 1101001

1110 1110100

1111 1111111

The set of code words in Table 7-1 may be generated from the set of messages by means of a matrix 

modulo-2 multiplication: 

v = uG

(7-1) 

where v = a code word (v7, v6, v5, v4, v3, v2, v1, v0) 

u = the corresponding message (u3, u2, u1, u0) 

G = the generator matrix

























0 0 0 1 0 1 1

0 0 1 0 1 1 0

0 1 0 0 1 1 1

1 0 0 0 1 0 1

It may be confirmed by inspection that every pair of code words in Table 7-1 differs in at least three bit 

positions, so d = 3 for this code. 

7-4 
 JSC-CR-06-072 

7.5.2 BER Calculation for Block Coding FEC 

The output BER for a block FEC decoder depends on the number of bit errors per code word that the 

code can correct. This quantity is related to the minimum distance for the code as follows: 





 



 − = 2

1 Int d

t

(7-2) 

where t = number of correctable bit errors per code word 

 Int(x) = function that truncates x to an integer

In the (7, 4) linear block code example in Table 7-1, the minimum distance is 3, so the code can correct 

one bit error per code word. 

For a hard-decision decoder, calculating the output BER is a two-step process. 

The first step is to calculate the probability of a code-word error, which is also called the word error rate

(WER). This depends on the input BER and the parameters n, k, and t (which depends on d). If it is 

assumed that input bit errors are randomly distributed, and that the bit errors are statistically independent 

events, then the WER can be determined with a simple equation. The code word will be correct if it has 

from 0 to t bit errors (because the decoder can correct that many errors). Consequently, the WER is: 

∑=

− − − = −

t

j

j n j BER BER

n j j

n WER

0

in in (1 ) ( )! !

! 1

(7-3) 

where BERin = input BER 

The second step, assuming a code word error has been made, is to estimate the number of erroneous bits 

in the output message. Assuming that all of the possible erroneous messages are equally likely, the 

output BER is (Reference 4-2): 

BER WER k

k

2 1

2 1

out − =

−

(7-4) 

The results of applying Equations 7-3 and 7-4 to the (7, 4) code example in Table 7-1 (with t = 1) are 

shown in Table 7-2. 

7-5 
JSC-CR-06-072 

Table 7-2. Performance of (7, 4) Linear Block Code With t = 1 

Input BER Output BER Improvement 

0.1 0.08 Negligible

0.01 0.0011 Order of magnitude 

0.001 0.000011 Two orders of magnitude 

For soft-decision block decoders, the decoding algorithms do not yield simple equations such as 

Equation 7-3. However, simulations show that soft-decision systems are about 2 dB better than harddecision systems; i.e., Eb/No must be about 2 dB greater in a hard-decision system to get the same BER 

as in the corresponding soft-decision system. 

As the above example illustrates, to calculate the output BER from the input BER for a hard-decision 

decoder, it is necessary to know the following: 

•

•

•

•

•

•

The number of bits per code word (n) 

The number of bits per message (k) 

The number of bits per code word that can be corrected (t) 

A wide variety of FEC codes are available. These are typically specified by name and by the above 

parameters. 

7.5.3 Bose-Chaudhurie-Hocquenghem Codes 

An important class of block codes is the class of Bose-Chaudhurie-Hocquenghem (BCH) codes. These 

codes have the following properties: 

= 2 −1 m n , where m is an integer greater than 2

d ≥ 2t +1

r = n − k ≤ mt , where r is the number of redundancy bits in a code word 

7-6 
 JSC-CR-06-072 

The parameters for several BCH codes are listed in Table 7-3. A much more complete table with n

extending to 1023 that can be used to obtain the number of correctable bit errors t is available in the 

literature.7-1

Table 7-3. BCH Code Parameters for n ≤ 63 

n k t

7 4 1

15 11 1

7 2

5 3

31 26 1

21 2

16 3

11 5

6 7

63 57 1

51 2

45 3

39 4

36 5

30 6

24 7

18 10

16 11

10 13

7 15

Figure 7-1 shows the output BER vs. input BER curve for a hard-decision (255, 239) BCH decoder, 

which is implemented in some communications systems. This decoder has r = 16, m = 8, and t = 2. The 

curve was generated by applying Equations 7-3 and 7-4. 

 

7-1 Shu Lin and Daniel Costello, Error Control Coding: Fundamentals and Applications, Prentice-Hall, Inc., 

Englewood Cliffs, NJ, 1983. 

7-7 
JSC-CR-06-072 

0.00

0.00

0.00

0.00

0.01

0.10

0.00001 0.0001 0.001 0.01

Input BER

Output B

E

R

10-3

10-4

10-5

10-6

10-1

10-2

10-4 10-5 10-3 10-2

Figure 7-1. Output BER vs. Input BER Curve for Hard-decision (255, 239) BCH Decoder 

7-8 
 JSC-CR-06-072 

7.5.4 Hamming Code 

The Hamming code is a relatively simple single-error correcting code. It is a special case of the BCH 

code with r = m . Therefore, it has the following properties: 

•

•

•

= 2 −1 r n , where r is the number of redundancy bits in a code word 

d = 3 

t = 1 

The (7, 4) Hamming code is sometimes extended to (8, 4) by the addition of another redundancy bit. 

This does not affect the number of correctable errors, but it is more convenient for implementing a 

system with the code rate R = 1/ 2 . 

Figure 7-2 shows the output BER vs. input BER curve for a hard-decision extended (8, 4) Hamming 

decoder. This decoder has r = m = 3. The curve was generated by applying Equations 7-3 and 7-4. 

Figure 7-3 shows BER vs. Eb/No curves for a soft-decision extended (8, 4) Hamming decoder and for 

various demodulator types. For these curves, the demodulator and FEC decoder are treated as one 

combined module. 

7.5.5 Hadamard Code 

This (n, k) block code is based on the Hadamard matrix, which is an n× n matrix with the property that

all of the rows (except one) have an equal number of 1-bits and 0-bits. The code words are simply the 

rows of the matrix. Other properties of the Hadamard code are 

•

•

•

k

n = 2

d = n / 2

t = Int[(d −1)/ 2] = Int[(n − 2)/ 4]

A typical example is the (64, 6) Hadamard code, which can correct up to 15 bits per code word. This 

capability is made possible by the 58 bits of redundancy that are added to each 6-bit message. 

Hadamard matrices may be used to define code sets for direct-sequence spread-spectrum systems. The 

equal-weight property is desirable for this application. 

7-9 
JSC-CR-06-072 

0.00

0.00

0.00

0.00

0.01

0.10

0.0001 0.001 0.01 0.1

Input BER

Output B

E

R

10-3

10-4

10-5

10-6

10-1

10-2

10-3 10-4 10-2 10-1

Figure 7-2. Output BER vs. Input BER Curve for Hard-decision (8, 4) Hamming Decoder 

7-10 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-2 0246 8 10 12

Eb/No (dB)

BE

R

14

10-1

10-2

10-3

10-4

10-5

10-6

DPSK

PSK

MSK

CFSK

NCFSK

Figure 7-3. BER vs. Eb/No Curves for Soft-decision (8, 4) Hamming Decoder 

7-11 
JSC-CR-06-072 

Figure 7-4 shows the output BER vs. input BER curve for a hard-decision (64, 6) Hadamard decoder. 

The curve was generated by applying Equations 7-3 and 7-4. Figure 7-5 shows BER vs. Eb/No curves 

for a soft-decision (64, 6) Hadamard decoder and for various demodulator types. For these curves, the 

demodulator and FEC decoder are treated as one combined module. 

7.5.6 Golay Code 

The Golay code is an (n, k) block code with the following properties: 

•

•

•

•

•

•

n = 23

k = 12

d = 7

t = 3

The Golay code is sometimes extended to (24, 12) by the addition of another redundancy bit. This does 

not affect the number of correctable errors, but it is more convenient for implementing a system with the 

code rate R = 1/ 2 . 

Figure 7-6 shows the output BER vs. input BER curve for a hard-decision extended (24, 12) Golay 

decoder. The curve was generated by applying Equations 7-3 and 7-4. Figure 7-7 shows BER vs. Eb/No

curves for a soft-decision extended (24, 12) Golay decoder and for various demodulator types. For these 

curves, the demodulator and FEC decoder are treated as one combined module. 

7.5.7 Reed-Solomon Codes 

In a Reed-Solomon (RS) encoder, the input bits are grouped to form RS symbols (not to be confused 

with M-ary demodulator symbols). Then the RS symbols are grouped into messages, and redundancy 

RS symbols are added to form code words. The code word length and the message length are expressed 

in terms of RS symbols rather than bits. RS codes have the following properties: 

= 2 −1 m n , where n is the number of RS symbols per code word and m is the number of bits per 

RS symbol 

d = n − k +1, where d is the minimum distance measured in RS symbols and k is the number of

RS symbols per message 

7-12 
 JSC-CR-06-072 

0.00

0.00

0.00

0.00

0.01

0.10

0.01 0.1 1

Input BER

Output B

E

R

10-1

10-2

10-3

10-4

10-5

10-6

10-1 10-2 100

Figure 7-4. Output BER vs. Input BER Curve for Hard-decision (64, 6) Hadamard Decoder 

7-13 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-4 -2 0 2468 10 12 1

Eb/No (dB)

BE

R

4

10-1

10-2

10-3

10-4

10-5

10-6

PSK DPSK

MSK

CFSK

NCFSK

Figure 7-5. BER vs. Eb/No Curves for Soft-decision (64, 6) Hadamard Decoder 

7-14 
 JSC-CR-06-072 

0.00

0.00

0.00

0.00

0.01

0.10

0.001 0.01 0.1

Input BER

Output B

E

R

10-1

10-2

10-3

10-4

10-5

10-6

10-1 10-2 10-3

Figure 7-6. Output BER vs. Input BER Curve for Hard-decision (24, 12) Golay Decoder 

7-15 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

-2 024 6 8 10

Eb/No (dB)

BE

R

12

10-1

10-2

10-3

10-4

10-5

10-6

DPSK

PSK

MSK

CFSK

NCFSK

Figure 7-7. BER vs. Eb/No Curves for Soft-decision (24, 12) Golay Decoder 

7-16 
 JSC-CR-06-072 

Equation 7-2 can still be used for RS codes, but d and t are expressed as numbers of RS symbols rather 

than bits. For example, a (15, 13) RS code has 4 bits per RS symbol, 13 RS symbols per message, and 

15 RS symbols per code word. The minimum distance for the (15, 13) RS code is 3 RS symbols and the 

number of correctable RS symbols in a code word is 1. 

The symbol-based structure of RS codes can be used to protect a system from burst errors, which are 

often caused by intermittent interference. For example, a (15, 13) RS code can correct one symbol per 

15-symbol code word, regardless of the number of erroneous bits in the erroneous symbol. 

Because a given RS symbol is formed by grouping m bits, it is correct only if all m bits are correct. 

Therefore, the RS SER is given by: 

m SER 1 (1 BER ) in = − − in

(7-5) 

where SERin = RS symbol error rate at RS decoder input 

BERin = bit error rate at RS decoder input 

Because the RS decoder operates on RS symbols rather than bits, Equation 7-3 is modified to show the 

dependency on RS SER (Reference 4-2): 

∑=

− − − = −

t

j

j n j SER SER

n j j

n WER

0

in in (1 ) ( )! !

! 1

(7-6) 

Figure 7-8 shows the output BER vs. input BER curve for a hard-decision (255, 223) RS decoder. In 

this case, m = 8 and t = 16. The curve was generated by applying Equations 7-5, 7-6, and 7-4. 

7.5.8 Interleaving

Every FEC block code has the ability to correct t erroneous bits per n-bit code word. If a code word has 

more than t erroneous bits, the code word is decoded incorrectly. Such a code is very effective at 

protecting against the occasional random errors that result from noise and noiselike interference. 

Intermittent interference, which can be modeled as a sequence of pulses, is another matter. Intermittent 

interference often results in a very high bit-error probability during a pulse. If the pulse duration Ti is 

long enough that it can corrupt more than t bits in a particular codeword, the FEC decoder will be unable 

to correct the errors. This will occur when T , a condition referred to as overloading. i b > tT

7-17 
JSC-CR-06-072 

0.00

0.00

0.00

0.00

0.01

0.10

0.001 0.01

Input BER

Output B

E

R

10-3

10-4

10-5

10-6

10-1

10-2

10-3 10-2

Figure 7-8. Output BER vs. Input BER Curve for Hard-decision (255, 223) RS Decoder 

7-18 
 JSC-CR-06-072 

Figures 7-9 and 7-10 illustrate pulsed interference to a digital system. Figure 7-9 illustrates overloading 

for a sequence of four code words (W1, W2, W3, W4), assumed to be part of a longer transmission for a 

system using a (7, 4) code. This code can correct one bit error per code word. Because the interference 

pulse is longer than one bit, the decoder is overloaded. 

7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1

W4 W3 W2 W1

The interference pulse

is longer than t bits.

tTb Ti

Figure 7-9. Pulsed Interference Without Interleaving 

(a)

(b)

(c)

7 7 7 7 6 6 6 6 5 5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1

7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1

Interleaving ensures that, in the

channel, consecutive bits are not

from consecutive code words.

7 7 7 7 6 6 6 6 5 5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1

The interference affects two bits, but

they are from different code words.

7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1

W4 W3 W2 W1

The interference is effectively split into

two small pulses that are shorter than Tb

.

Figure 7-10. Pulsed Interference With Interleaving 

7-19 
JSC-CR-06-072 

Interleaving the source data prior to transmission addresses this problem. Interleaving reorganizes the 

source data, so that consecutive bits in the channel are not from the same code word. With interleaving, 

bits corrupted by an interference pulse are distributed across multiple code words. 

Figure 7-10 is for the same example as in Figure 7-9, but with interleaving. The interleaving process is 

illustrated in Figure 7-10(a). In Figure 7-10(b), the interference pulse is shown corrupting two bits, as 

before. When the data is demodulated and de-interleaved in the receiver, however, the corrupted bits are 

seen to reside in two different code words. This is shown in Figure 7-10(c). The effect is the same as 

splitting the interference pulse into two shorter – and therefore less harmful – pulses. 

In this example, the interleaving process is applied to a group of four codewords. The process would be 

repeated for every group of four code words for the duration of the transmission. By increasing the 

number of code words in the interleave group (referred to as the interleaver depth L), the protection is 

extended to longer pulses. When the data is interleaved with depth L, an interference pulse will 

overload the decoder only if T . i b > LtT

7.6 CONVOLUTIONAL CODES 

7.6.1 Description 

Convolutional coding is the other major approach to FEC coding. Convolutional coding and decoding 

techniques are very briefly described here. A more detailed discussion of convolutonal codes can be 

found in the literature (for example, Reference 4-2). 

In contrast to block coding, which transforms k-bit messages into n-bit code words, convolutional 

coding is applied to a continuous stream of data to produce another continuous stream of data. 

Redundancy is added in the process, so the encoder output data rate is higher than the encoder input 

(information) data rate. The ratio of input to output data rates is called the rate of the code, denoted R. 

Convolutional encoding is implemented with a shift register of several stages, and two or more 

modulo-2 adders. The number of stages is called the constraint length of the code, denoted k. Certain 

stages are connected to the adders, so that at any given time the output of each adder is the modulo-2 

sum of the shift register stages to which it is connected. The input to the shift register is the sequence of 

information bits, so the clock rate of the shift register is the data rate of the information sequence. The 

output is taken from the adders. Each adder is sampled in turn, such that all of the adders are sampled 

once per clock cycle. 

7-20 
 JSC-CR-06-072 

Convolutional codes are given the designation (R, k), in recognition of the dependence on the rate and 

constraint length. In addition to these parameters, a particular convolutional code is characterized by the 

specific connections between shift register stages and adders. These connections may be designated 

graphically, using a diagram of the sequential logic circuit. Alternatively, the connections may be 

specified in terms of code tap connections. A code tap connection is represented by a k-bit symbol 

( , , , ) that designates the connections of a particular adder. If the i G = gk−1 K g1 g0

gi = 1

th stage is connected to 

the adder, then , otherwise . There will be one code tap connection symbol for each adder 

in the encoder circuit. A (1/2, 3) convolutional encoder is illustrated in Figure 7-11. 

gi = 0

+

+

G1 = 111

G2 = 101

a

b Input Output

... i

3 i2 i1 i0 ... b3 a3 b2 a2 b1 a1 b0 a0

Figure 7-11. A (1/2, 3) Convolutional Encoder 

Several convolutional codes are listed in Table 7-4. These are all rate-1/2 codes, so there are two code 

tap connections for each code. 

Table 7-4. Rate R = 1/2 Convolutional Codes 

Constraint Length

k

Code Tap Connections

G1, G2

3 111, 101 

4 1111, 1101 

5 11101, 10011

6 111101, 101011 

7 1111001, 1011011 

8 11111001, 10100111 

7-21 
JSC-CR-06-072 

7.6.2 Decoding Convolutional-Coded Data 

The convolutional decoder examines an incoming sequence of data to determine the likelihood that such 

a sequence could have actually been transmitted. If the decoder decides that the sequence contains

errors, it makes the necessary adjustments to the data. Unlike the block decoder, the convolutional 

decoder is not limited to a particular sequence length. Typically, convolutional decoders examine 

sequences that are several times longer than the constraint length. This process is called finite-memory 

decoding. 

The decoding process is based on the state-transition properties of the encoder, which ensures that only 

certain coded sequences are possible. A convolutional code can be represented by a code tree, which is a 

diagram that indicates the transitions – and therefore the sequences – that are possible. One of the most 

important and universal convolutional decoders is called the Viterbi algorithm. This algorithm is 

typically applied to convolutional codes with rates of 1⁄2 to 3⁄4, constraint lengths of 3 to 8, and memories 

of 3 or 4 times the constraint length. Viterbi decoders generally employ soft-decision decoding. 

7.6.3 BER Calculation for Convolutional Coding 

There is an analytical approach for estimating the output BER for a convolutional decoder. An 

explanation of the technique and of the parameters involved is beyond the scope of this handbook. The 

objective is to simply show how the BER can be calculated. 

The technique involves two parameters D and N and three functions T(D), T(D,N), and P(D). The 

transfer function T(D) of the convolutional code is derived from the state-transition representation of the 

code. It is a polynomial: 

∑

∞

=

=

j d

j T(D) a jD

(7-5) 

where d = minimum free distance of the code 

 aj = polynomial coefficient for the code 

The transfer function is extended to form T(D,N) by adding a parameter N, which is also based on the 

state-transition representation. Then that function is differentiated with respect to N, and evaluated at 

N = 1. The new polynomial function P(D) may be called the derivative polynomial: 

7-22 
 JSC-CR-06-072 

∑

∞

= =

= =

j d

j

j

N

T D N c D

dN

d P D

1

( ) ( , )

(7-6) 

where cj = derivative polynomial coefficient for the code 

Lists of the functions T(D) and P(D) for various codes are published in the literature, and are often 

provided with system descriptions. A few examples are given in Tables 7-5 and 7-6. 

Table 7-5. Transfer Functions for R = 1/2 Convolutional Codes 

k T(D) 

3 D5 + 2D6 + 4D7 + 8D8 +16D9 L

4 D6 + 3D7 + 5D8 +11D9 + 25D10 L

5 2D7 + 3D8 + 4D9 +16D10 + 37D11L

6 D8 + 8D9 + 7D10 +12D11 + 48D12 L

7 11D10 + 38D12 +193D14 +1331D16 + 7230D18L

Table 7-6. Derivative Polynomials for R= 1/2 Convolutional Codes 

k P(D) 

3 D5 + 4D6 +12D7 + 32D8 + 80D9 L

4 2D6 + 7D7 +18D8 + 49D9 +130D10 L

5 4D7 +12D8 + 20D9 + 72D10 + 225D11L

6 2D8 + 36D9 + 32D10 + 62D11 + 332D12 L

7 36D10 + 211D12 +1404D14 +11633D16 + 76628D18L

The first step is to identify the derivative polynomial P(D) for the code. The lowest degree is d, the 

minimum free distance of the code. The next step is to identify the hard-decision demodulator transfer 

function. This equation expresses the BER at the demodulator output as a function of Eb/No: 

7-23 
JSC-CR-06-072 















 =

o

hard N

E

BER F b

(7-7) 

where BER hard = BER for a hard-decision demodulator 

F(x) = function for a given demodulator type 

The BER at the decoder output is approximated by the following upper bound: 















 = ∑

∞

= No

E

BER c F jR b

j d

j

(7-8) 

where R = code rate of the convolutional code 

As an example, consider a BPSK system with soft-decision (1/2, 7) Viterbi decoding. The derivative 

polynomial P(D) is the last one listed in Table 7-6. Equation 6-17 gives the BPSK function F(x):

F x erfc( x ) 2

1 ( ) =

(7-9) 

Substituting that function into Equation 7-8 gives:



 +L













+















+













 =















 = ∑

∞

=

o o o

o

2

14 erfc

2

1404

2

12 erfc

2

211

2

10 erfc

2

36

erfc

2

1

N

E

N

E

N

E

N

E BER c jR

b b b

b

j d

j

(7-10) 

Of the five P(D) terms shown in Table 7-6, only three are explicitly shown in Equation 7-10. More

terms can be included for greater accuracy. 

These equations apply to soft-decision decoding. If hard-decision convolutional decoding is 

encountered (which is unlikely), its performance will be roughly 2 dB worse. 

7-24 
 JSC-CR-06-072 

7.6.4 BER Curves 

Figure 7-12 shows the BER vs. Eb/No curves for a soft-decision (1/2, 7) Viterbi decoder and for various 

demodulator types. For these curves, the demodulator and FEC decoder are treated as one combined 

module. 

7.7 CPM 

7.7.1 Description 

The term continuous-phase modulation (CPM) refers to signals that are designed to maintain continuity 

of phase for the duration of the transmission. This constraint results in a phase-modulated or frequencymodulated signal that has memory. This memory can be used to improve the BER of the demodulated 

signal, but at the cost of an increase in receiver complexity. The complexity is due to the fact that the 

receiver must perform recursive processing of the received signal to exploit the memory. One approach 

is to employ convolutional encoding and decoding in addition to the CPM. The receiver can incorporate 

the Viterbi decoding algorithm into the demodulation process. This algorithm is designed to consider 

signal history when determining the most probable value of the current demodulated symbol. 

CPM generally encompasses a broad class of modulation types. The curves presented here are for 

continuous-phase FSK signals with Viterbi decoding. The modulation index is the ratio of the spacing 

of adjacent tones to the symbol rate 

7.7.2 BER Curves 

Figure 7-13 shows the BER vs. Eb/No curves for a soft-decision CPM receiver with M = 2 and a 

modulation index of 1/2. Figure 7-14 shows the BER vs. Eb/No curves for a soft-decision CPM receiver 

with M = 4 and a modulation index of 1/6. For these curves, the demodulator and FEC (Viterbi) decoder 

are treated as one combined module.

In these graphs, the term “interference” and the variable I refer to CW interference. Any noise-like 

interference power is simply assumed to be added (in mW) to the noise power. For example, suppose 

there are two interfering signals: a CW signal and a noise-like signal. The S/I parameter is the ratio of

the desired signal power to the CW interfering signal power, and the x-axis variable Eb/No is calculated 

with Equation 6-4, where N is the total noise-like power (including the receiver noise and the noise-like 

interfering signal). 

7-25 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

0 1 2345 6 7 8 9 1

Eb/No (dB)

BE

R

0

10-1

10-2

10-3

10-4

10-5

10-6

DPSK

PSK CFSK

NCFSK

Figure 7-12. BER vs. Eb/No Curves for Soft-decision (1/2, 7) Viterbi Decoder 

7-26 
 JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

0 4 8 12 16

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB

S/ I = 6 dB

No interference

Figure 7-13. BER vs. Eb/No Curves for Binary CPM Decoder (Modulation Index 1/2) 

7-27 
JSC-CR-06-072 

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

2 6 10 14 18 22 26

Eb/No (dB)

BE

R

10-1

10-2

10-3

10-4

10-5

10-6

S/ I = 10 dB No interference

Figure 7-14. BER vs. Eb/No Curves for 4-ary CPM Decoder (Modulation Index 1/6) 

7-28 
 JSC-CR-06-072 

The curves were generated by simulation. Each figure displays multiple curves. Each curve is a plot of

BER vs. Eb/No. The curve labeled “No interference” applies to the case in which there is no CW

interference. The other curves are for cases with CW interference. Each of those curves is labeled with 

the S/I for that curve. As expected, each curve shows that the BER decreases as the Eb/No increases. As 

also expected, for a given Eb/No the BER decreases as the S/I increases. 

7.8 CONCATENATED CODES 

An RF communications system may employ two FEC coding systems to increase the protection against 

errors. The two codes are referred to as the inner code and the outer code, as shown in Figure 7-15. A 

typical application is a system that must guard against both random and burst errors. Suppose an (n, k) 

RS code capable of correcting two symbol errors per code word is specified as the outer code. If an

interference burst can cause at most two symbol errors, the RS decoder can correct those errors. 

However, if in addition to those two errors there is another (random) error somewhere else in the code 

word, the decoder will be overloaded. The probability of random (non burst-related) errors occurring 

can be reduced by incorporating an inner code. This code would not be able to correct errors due to 

bursts, leaving that task to the outer code. The inner code would simply minimize the BER between 

bursts. A convolutional code is often used as the inner code in a concatenated coding system. 

data source FEC

encoder

FEC

encoder

digital

modulator

channel

output FEC

decoder

FEC

decoder

digital

demod

Outer

Code

Inner

Code

Figure 7-15. Concatenated Codes 

The inner decoder may be either a soft-decision or hard-decision device, and the outer decoder is a harddecision device. To analyze a concatenated coding scheme, the transfer functions for the individual 

decoders are computed and then combined. For example, if the inner decoder is a soft-decision decoder, 

then the demodulator and inner decoder are treated as one combined module. For a given Eb/No and S/I, 

7-29 
JSC-CR-06-072 

the output BER for that combined module is obtained from the appropriate curve. Then viewing that 

BER value as the input BER for the outer decoder, the output BER for the outer decoder is obtained

from the curve for that device. 

7-30 
 JSC-CR-06-072 

SECTION 8 – SOURCE DECODER 

In a transmitter, the source encoder converts the source information into a sequence of bits at a certain 

data rate. If the source is analog, an ADC samples and encodes the analog waveform. If the source is 

digital, the source encoder reformats and retimes the input data if necessary. The output of the source 

encoder is a sequence of information bits, at the information data rate. 

In the receiver, the source decoder converts the information bit sequence to the format required by the 

user of the receiver. If this format is analog, a DAC will be employed at this stage. The output of the 

source decoder is the output of the receiver. In general, it will differ somewhat from the original 

information signal due to noise and interference. This difference can be quantified, and is a measure of 

the overall system performance. 

A DAC, along with its associated ADC, is used in a hybrid communications system. A hybrid system

uses digital processing and transmission techniques to communicate analog information. When used to 

transmit speech, hybrid systems usually provide duplex capability; in such an application the ADC/DAC 

pair are often combined in a single unit referred to as a codec. There are various techniques available 

for performing the analog-to-digital and digital-to-analog conversion; the particular technique employed 

has an effect on the fidelity of the reconstructed analog signal as well as the sensitivity of the system to 

channel-induced bit errors. 

8.1 MULTI-BIT SAMPLING 

8.1.1 Description 

Pulse-code modulation (PCM) is a straightforward conversion algorithm. The analog waveform is 

sampled at regularly spaced intervals established by the sampling frequency, and each sample is coded 

into a k-bit digital word according to its amplitude. Because there are multiple bits associated with each 

sample, PCM is also called multi-bit sampling. The analog waveform is thus represented by a sequence 

of k-bit digital words. The resulting bit stream is transmitted; channel impairments typically result in a 

number of errors in the received demodulated bit stream. In the receiver, the DAC decodes the digital 

words and produces a sequence of samples. Signal conditioning operations such as interpolation and 

lowpass filtering are then performed on the samples, resulting in a reconstructed analog waveform that 

should be a close approximation to the original information signal. The block diagram for a PCM ADC 

and DAC appears in Figure 8-1. 

8-1 
JSC-CR-06-072 

Sample

Quantize

Code

Transmitter

processing

Channel

Signal

conditioning

Decode

Receiver

processing

Initial waveform

x(t)

Final waveform

y(t) Samples

x(nT)

Quantized samples

x

q(nT)

Bits

bT

Bits

bR

Quantized samples

yq(nT)

ADC

DAC

Figure 8-1. Block Diagram for PCM ADC and DAC 

The input signal to the transmitter is denoted x(t) in Figure 8-1. The Sample operation produces a 

sample sequence denoted x(nT), where T is the sampling interval and n = 1, 2, ..., L. These samples are 

then quantized. For k-bit PCM, the number of quantization levels is 2k

. Each sample x(nT) is replaced 

by the nearest quantization level xq(nT) . The Code operation then replaces each quantized sample with 

a sequence of k bits. For example, in a 6-bit ADC there are 26

 = 64 quantization levels. These levels in 

decimal form are 0, 1, 2, ..., 63, where the 0 value represents the minimum signal and the 63 value 

8-2 
 JSC-CR-06-072 

represents the maximum signal. If a particular sample is quantized to the decimal level 5, then the Code 

operation replaces the decimal level 5 by its binary form 000101. Thus, a sequence of L samples 

becomes a sequence bT of kL bits. 

Then the bit sequence is processed in the transmitter to create the channel waveform. This processing 

includes modulation and perhaps error correction coding and spreading. The channel waveform travels 

across the channel to the receiver, where it is processed to recover the received bit sequence bR. The 

receiver processing includes demodulation and perhaps error correction decoding and despreading. If 

the transmission is error-free, then bR = bT. 

When the received bit stream is decoded, the result is again a sequence of quantized samples. This

sequence is denoted yq(nT) in Figure 8-1. If bR = bT, then yq(nT) = xq(nT). Then the sequence of 

samples is conditioned (e.g., interpolated and lowpass filtered) to convert it to a continuous analog 

signal, denoted y(t) in Figure 8-1. In good conditions, y(t) ≈ x(t). 

Note that even in the absence of bit errors, the DAC signal conditioning block may not reproduce x(t) 

with perfect fidelity, because of information loss in the quantization process. The resulting distortion, 

i.e., the difference between x(t) and y(t) in error-free transmission, is referred to as quantization noise

and is a performance-limiting factor in any hybrid communications system. It should be noted that 

quantization noise can be made arbitrarily small by increasing the number of bits per sample. The tradeoff, of course, is an increase in data rate and required bandwidth. 

8.1.2 Output Signal-to-Noise 

A fundamental assumption in a performance degradation analysis of hybrid communications systems is 

that the noise due to quantization and the noise due to channel-induced bit errors are additive. It is

appropriate, therefore, to specify a source decoder transfer function that gives output S/N (including 

both types of noise) as a function of input BER. 

The source decoder performance curves for a multi-bit (PCM) decoder 8-1

 are given in Figure 8-2. The 

S/N vs. BER relationship is shown for k-bit PCM with k = 4, 5, 6, 7, and 8. The sampling rate in each 

case is 8 kHz. 

 

8-1 L. Rabiner and R. Schaefer, Digital Processing of Speech Signals, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1978. 

8-3 
JSC-CR-06-072 

0

10

20

30

40

-6 -5 -4 -3 -2 -1

Input BER

Output S/N (

d

B)

4-bit

5-bit

6-bit

7-bit

8-bit

10-1 10-2 10-3 10-4 10-5 10-6

Figure 8-2. Output S/N vs. Input BER for PCM DAC 

8-4 
 JSC-CR-06-072 

8.2 SINGLE-BIT SAMPLING 

8.2.1 Description 

Differential PCM (DPCM) is a signal conversion scheme similar to PCM in that the ADC samples an 

analog waveform. Rather than sampling, quantizing, and coding the actual input waveform, the ADC 

performs these operations on the difference between the waveform and an estimate of the waveform. A 

DPCM encoder is shown in Figure 8-3. The input waveform, estimate of the waveform, and difference 

signal are denoted x(t), xE(t), and d(t), respectively. The estimation algorithm is designed so that the 

estimate xE(t) closely resembles the actual waveform x(t). The difference signal d(t), therefore, tends to 

be relatively low in amplitude. Consequently, the difference signal can be coded with relatively few bits 

per sample, resulting in a more efficient system than strict PCM coding. Also, the estimation algorithm

itself can serve as the receiver DAC. 

Perhaps the simplest DPCM estimation algorithm involves a single-sample delay. The underlying 

assumption is that the input waveform varies slowly enough so that the present sample is very close to 

the previous sample. Delta modulation (DM) is an example of such a technique, with the further 

simplifying assumption that the difference signal can be adequately represented by a one-bit code. In 

other words, a DM ADC simply transmits a single bit according to whether the input waveform sample 

has made a positive or negative excursion with respect to the previous sample. If no difference is

detected, the system will typically idle, transmitting a sequence of alternating 1-bits and 0-bits. This 

technique is also called single-bit sampling or oversampling (because the sampling rate must be high to 

ensure that the waveform varies slowly between samples). 

Signal conversion algorithms involving estimation techniques, such as DPCM, often permit the 

estimation procedure the flexibility to adapt in response to certain conditions. If the DPCM difference 

signal exceeds some specified value, for example, the estimation procedure may change in such a way as 

to reduce the difference in the future. This technique has been successfully applied to DM systems 

resulting in what are known as adaptive DM (ADM) systems. One variation of ADM that is currently 

popular, owing to its simplicity and resistance to channel errors, is continuously variable slope delta 

(CVSD) modulation. CVSD systems are capable of adequate performance at bit error rates as high as 

10%. For this reason, CVSD systems are appropriate for use in communications applications where 

8-5 
JSC-CR-06-072 

systems must function properly under adverse channel conditions. The CVSD algorithm and the other 

signal conversion techniques referred to in this section are described more thoroughly in many texts.8-2 

Sample Quantize Code

Estimation

Algorithm

+

x(t)

-xE(t)

d(t) d(nT) d

q

(nT) bT

Figure 8-3. Block Diagram for DPCM Encoder 

8.2.2 Output Signal-to-Noise 

Figure 8-4 shows a BER vs. S/N curve for 16-kbit CVSD.8-3 Comparing the curve with the curves in 

Figure 8-2, note that 6-, 7-, and 8-bit PCM outperform CVSD at low values of BER, but CVSD 

outperforms PCM when the BER is large. Note that CVSD achieves this performance at a data rate 

between one-fourth and one-third the data rate of the comparable PCM system. 

 

8-2 N. Jayant and P. Noll, Digital Coding of Waveforms, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984. 8-3 E. Harras and J. Preusse, Communication Performance of CVSD at 16/32 Kilobits/second, CommunicationsElectronics System Integration Office, US Army Electronics Command, Fort Monmouth, NJ, 1974.

8-6 
 JSC-CR-06-072 

0

10

20

30

40

-6 -5 -4 -3 -2 -1

Input BER

Output S/N (

d

B)

10-1 10-2 10-3 10-4 10-5 10-6

Figure 8-4. Output S/N vs. Input BER for 16-kbit CVSD 

8-7/8-8 
 JSC-CR-06-072 

SECTION 9 – SAMPLE RECEIVER ANALYSIS PROBLEMS 

This section presents several examples of using this handbook to perform a receiver analysis. 

Section 2.4.5 lists the specific steps required to perform an analysis. 

9.1 RECEIVER PERFORMANCE ANALYSIS EXAMPLE 1 

A two-way radio system is being planned to support digital (speech) communications. BPSK 

modulation will be used. There is some flexibility in the system design regarding transmitter power and 

whether or not FEC is used. The S/N of the output voice signal is required to be greater than 15 dB. 

An RF analysis is required to 

•

•

•

•

•

•

•

•

•

•

•

Estimate input signal requirements without FEC 

Estimate input signal requirements with FEC 

Compare the two implementations 

9.1.1 Approach 

The following approach will be taken: 

Model each system

Work backwards for each system to calculate input signal requirement 

Compare the two implementations 

9.1.2 Models 

The characteristics of System 1 are as follows: 

DAC: 48 kBit PCM, 8000 samples/s at 6 bits/sample 

FEC: none 

Demodulator: BPSK 

IF Bandwidth: 100 kHz 

The characteristics of System 2 are as follows: 

DAC: 48 kBit PCM, 8000 samples/s at 6 bits/sample 

9-1 
JSC-CR-06-072 

•

•

•

•

•

•

•

•

•

•

•

•

•

FEC: Extended Golay (24, 12) soft-decision 

Demodulator: BPSK 

IF Bandwidth: 200 kHz 

System 2 requires twice the bandwidth because the coded data rate is twice the uncoded data rate. 

9.1.3 Calculate Input Signal Requirement for System 1 

The following analysis steps apply for System 1: 

The receiver has the RF/IF section, demodulator, and source decoder modules. 

There is no interference – the analysis will consider only the effects of noise. 

The required performance measure is a minimum output S/N of 15 dB.

Using the Input BER vs. Output S/N curve for 48 kBit PCM (Figure 8-2), an output S/N = 15 

requires approximately BER = 10-3 at the input. 

Using the BPSK demodulator curve (Figure 6-2), an output BER = 10-3 requires an input Eb/No = 

7 dB. Using Equation 6-8 with a 100 kHz bandwidth and a 48 kbit/s bit rate, the corresponding 

S/N is 3.8 dB. 

9.1.4 Calculate Input Signal Requirement for System 2 

The following analysis steps apply for System 2: 

The receiver has the RF/IF section, demodulator, FEC decoder, and source decoder modules. 

There is no interference – the analysis will consider only the effects of noise. 

The required performance measure is a minimum output S/N of 15 dB.

Using the Input BER vs. Output S/N curve for 48 kBit PCM (Figure 8-2), an output S/N = 15 

requires approximately BER = 10-3 at the input. 

Using the BER vs. Eb/No curve for the soft-decision Golay decoder (Figure 7-7), for PSK an 

output BER = 10-3 requires an input Eb/No = 3 dB. Using Equation 6-8 with a 200 kHz 

bandwidth and a 96 kbit/s coded bit rate, the corresponding S/N is -0.2 dB. 

9.1.5 Comparing the Two Implementations 

System 2 requires 4 dB less S/N than System 1 to achieve the required output. However, the bandwidth 

of System 2 is twice that of System 1, so it will have twice the noise (i.e., 3 dB more). Thus, System 2 

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has a 1-dB power advantage. It can transmit slightly more power than System 1 and achieve the same

performance goal. If transmitting at the same power level, System 2 will have a slightly greater

communications range, at the expense of greater receiver complexity. 

9.2 RECEIVER PERFORMANCE ANALYSIS EXAMPLE 2 

This example has wider scope than the previous example. To illustrate the context in which a receiver 

analysis frequently occurs, a realistic problem is presented and solved with a combination of techniques, 

including techniques from Sections 3 and 5 of this handbook. 

A two-way FM voice radio system is being planned to support communication between a mobile radio 

and a fixed base station. The operational area extends to 100 km from the base station. The preferred 

operating frequency is at or near 160 MHz. The base station transmitter has 5 W of power and employs 

an omnidirectional antenna with 2 dBi of gain. The mobile antenna has not been specified, pending the 

outcome of the analysis. The system will be operated in an area with a moderate density of commercial 

and industrial development. In this area, it is known that there is a digital communications link 

operating in the same frequency band. 

Focusing on the performance of the mobile receiver, an RF analysis is required to 

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Validate the basic design decisions 

Determine the optimum mobile antenna 

Minimize the impact of the interfering digital communications link 

9.2.1 Approach 

The following approach will be taken: 

Summarize the operating environment 

Specify the equipment characteristics 

Specify or summarize the required performance goals 

Use link-budget analysis to estimate the received-signal power 

Utilize this handbook to estimate the system performance and answer the questions about the 

antenna and the interference. 

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9.2.2 Operating Environment 

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Size: Radius of mobility is 100 km, so the maximum distance for the desired link is 100,000 

meters.

Type: “Moderate density of commercial and industrial development” indicates the possibility of

signal fading. It will be assumed that previous measurements at a similar site determined that a 

20-dB fade margin is required. 

Interference: Digital (FSK) modulation, operating in a single 25-kHz channel in the VHF band, 

at 5 W.

9.2.3 Equipment Characteristics (for the Desired Signal) 

It is assumed that the following information has been determined: 

Transmitter power: 5 W = 5000 mW, 10 log(5000) = 37 dBm 

Base station antenna: omni, gain = 2 dBi 

Mobile antenna: unspecified 

Modulation: FM voice, frequency deviation ±5 kHz 

Receiver IF bandwidth: 16 kHz 

Receiver noise figure: 4 dB 

Adjacent Channel Rejection: 65 dB 

Frequency: VHF band, preferably at or near 160 MHz 

Receiver sensitivity: 0.3 μV for output S/N = 12 dB 

9.2.4 Required Performance 

The required performance is specified in terms of the AI. The interpretation of AI is subjective and is 

generally context-dependent. For this example, the following interpretation is assumed: 

Good: AI > 0.9 

Marginal: 0.7 < AI < 0.9 

Unacceptable: AI < 0.7 

Note: The receiver sensitivity can also be used to evaluate performance, but it does not address signal 

intelligibility and it does not address non-noise-like interference. 

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9.2.5 Received Signal Power Calculation 

Equation 2-1 is used to estimate the minimum received-signal power. The distance is assumed to be the 

maximum operating distance of 100 km. The mobile antenna is unspecified at this point, so its gain is 

arbitrarily set to 0 dBi. 

77 dBm

37 2 0 20 log (100,000) 20 log (160) 27.6

20 log ( ) 20 log ( ) 27.6

= −

= + + − − +

PR = PT + GT + GR − d − f +

9.2.6 Comparison of Received-Signal Power With Receiver Sensitivity

The receiver sensitivity is 0.3 uV (= 3 x 10-7 V). The following equation is used to convert this quantity 

to power: 

30 -117 dBm

50

9 10 10 log 30 10 log

2 14

+ = 













 ⋅ + = 













 =

−

R

V P rms

where 

P = signal power, in dBm

Vrms = RMS voltage of signal, in V 

R = antenna impedance (50 ohms) 

Thus a received-signal power of -117 dBm is required to produce an output S/N of 12 dB. The 

estimated minimum received-signal power is -77 dBm, which provides a considerable margin – more 

than enough to accommodate the 20-dB fade margin. This initial result is encouraging, but the 

handbook will be used to develop a more complete picture that includes the noise and interference. 

9.2.7 Noise Calculation 

The receiver noise power referenced to the receiver input is given by: 

( )

( )

128 dBm

4 144 10 log 16

144 10 log

= −

= − +

N = FR − + BR

where 

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FR = receiver noise figure, in dB 

BR = receiver IF bandwidth, in kHz 

In this equation, the term -144 dBm/kHz is the thermal noise power density at standard room

temperature (290 K). Depending on the frequency range of operation, it may be appropriate in some

cases to include an additional factor for man-made noise contributions. 

9.2.8 Interference Calculation 

The worst-case assumption is that the interference is on tune; i.e., in the desired-signal channel. The 

interfering signal bandwidth is about the same as the desired signal bandwidth since they both occupy a 

25-kHz channel. In this case, Section 3 indicates that there is no FDR. 

The undesired received-signal power can be estimated just like the desired signal by means of Equation 

2-1. However, both the desired and the undesired received-signal powers would have to be calculated at 

every point within the radius of mobility. The calculation of desired signal power that was done 

previously for maximum distance does not necessarily give the worst-case S/I. The S/I will vary as the 

mobile moves throughout the area. If the location of the interference were known, it would be possible 

to use propagation-modeling software to compute the area coverage in terms of S/I. Instead of 

performing those detailed calculations, the interference effects will simply be characterized here in 

general terms by using the handbook transfer functions. 

The relevant transfer function is Figure 5-6, which displays multiple AI vs. S/N curves. One curve is for 

the case with no interference. Each of the other curves is for a particular S/I value. Because the 

minimum received-signal power is -77 dBm and the noise power is -128 dBm, the worst-case S/N is 

(-77) - (-128) = 51 dB. Including the 20-dB fade margin, the worst-case S/N is 31 dB. 

It is clear from the plot that, in the absence of interference, this is more than enough S/N for proper 

operation. Since noise is not a problem, the operation is said to be interference-limited. The curves

show that an S/I of less than about 12 dB puts the system in the marginal range (AI < 0.9), and an S/I of 

less than about 6 dB puts the system in the unacceptable range (AI < 0.7). Because the desired and 

interfering transmitters have the same output power and both lie in the area of interest, there will be a 

region of points close to the interfering transmitter antenna at which the mobile receiver performance 

will be unacceptable (S/I < 6 dB). Propagation modeling is required to determine the size and shape of

the “denied” region. 

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9.2.9 Remediation 

There are several ways to remediate the problem:

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Increase the desired transmitter power. This may not be possible for practical, economic, or 

regulatory reasons. For example, it may degrade the performance of other systems in the 

environment. If it is possible, the action may shrink the denied region to an acceptable size. 

Employ a directional mobile antenna. Such an antenna would have to be steerable to ensure that 

it always pointed toward the desired antenna. However, there will be places where the mobile 

antenna cannot point toward the desired antenna without including the undesired antenna in the 

capture area. If the undesired antenna is directional and its pattern and orientation are known, it 

might be possible to operate in spite of being co-channel, but it would be difficult to perform the 

necessary geographic analysis. This action may also shrink the denied region to an acceptable 

size. 

Select a channel that takes the interference out of the receiver passband. This is the most 

practical and effective solution. Since the adjacent channel rejection is 65 dB, the handbook 

transfer functions show that the mobile receiver will then operate as required in most locations. 

(Because of nonlinear (cosite) interactions, there is usually a small denied region that cannot be 

eliminated.) The decision regarding the mobile antenna can then be based on other 

considerations. 

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