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### Chapter 7

Performance of Communication Systems Corrupted by Noise

Introduction(chapter objectives)

- Bit error rate for binary systems (unipolar, polar , bipolar, OOK, BPSK, FSK, and MSK)

- Output signal-to-noise ratio for analog systems(AM, SSB, PM, and FM)

- How much bandwidth occupied in the air?

How many AM stations can be?

- What is the listening-performance?

Is it very noisy to the listener?

- Equipment cost and complexity?

Bandwidth of Hz

SNR

Money and time

Introduction

Eample 2: A digital communication case:

- How much bandwidth occupied in the air?

How many AM stations can be?

- What is the listening-performance?

Is it very noisy to the listener?

- Equipment cost and complexity?

Data-rate of bsp or Bandwidth of Hz

Error rate (probability of error) or SNR

Money and time

Introduction(chapter objectives)

- Channel: not a “free lunch”
- Always having a limited bandwidth
- With noise, and even interference
- Often, with distortion

We mainly concern the most common AWGN channel:

- Additive noise
- Noise is White and Gaussian
- No distortion, maybe some amplifying of A, maybe some delay of t0
- Bandwidth is enough for a certain transmission

Introduction

In this chapter, we discuss the “performance-and-cost” of communication systems

- SNR and/or Pe (probability of error), bit-error-rate
- Bandwidth and/or data-rate

For digital system, bit-error-rate of:

OOK

BPSK

FSK

QPSK

MSK

For PCM system, Signal-to-Noise-Ratio

For analog system, Signal-to-Noise-Ratio of:

AM

SSB

PM

FM

Introduction

- Error probability for binary signaling
- General process and bit error rate
- Results for Gaussian Noise
- Optimal reception with Match-filters

Performance of baseband binary systems

Unipolar/Polar/Bipolar signaling

Coherent detection of bandpass binary system

Non-coherent detection of bandpass binary system

QPSK and MSK

Comparison of digital signaling systems

Output SNR for PCM systems

Output SNR for analog systems

Comparison of analog systems

7.1 Error Probabilities for Binary Signaling(General Results)

- Baseband signaling in Ch3, such as line-code:

processing = LPF+AMP

- Bandpass signaling in Ch4&5, as OOK, PSK, FSK, MSK, and etc. processing = superheterodyne receiver (mixer + IF-amp + detector)

7.1 Error Probabilities for Binary Signaling

Src data of 0 and 1 at prob. of ½ randomly

A signal, up to src data is in the interval (0,T)

Noise added

r(t) is then processed and r0(t) output.

Sampled at t0, the sync-timing; noise is inside of r0

r01 and r02 are diff, then which data could be told

Source data:

Transmitted signal

Input of receiver

After sampling

Data received

7.1 Error Probabilities for Binary Signaling

- Receiver: detection + decision

In the receiver, detect-processing is used to convert the waveforms r(t) into r0

- With no noise, s1(t) and s2(t) are mapped to the apart “clean centers”
- The noise corrupts the waveform and then the r0 is “dirty” and away from the center.

Decision is to obtain 0 or 1 from the r0

- The rule is “go to the nearest neighbor”
- Or equivalent to: use a threshold VT to seperate.

7.1 Error Probabilities for Binary Signaling

- Errors occur when r0 falls in the wrong areas due to noise.
- The r0 is a random variable, basically from the randomness of n(t) in r(t). The distribution could be as following:
- f(r0|s1) and f(r0|s2) are generated from n(t) by detect-processing
- VT is a proper-defined threshold

7.1 Error Probabilities for Binary Signaling

- When signal plus noise is present at the receiver input,Error can occur in two ways, an error occurs when r0<VT if a binary 1 is sent, and an error occurs when r0>VT if a binary 0 is sent

7.1 Error Probabilities for Binary Signaling

- P(s1 sent) and P(s2 sent) are source statistics and are known before transmission, so called prior statistics; P(s1 sent) and P(s2 sent) are normally considered to be equally likely.
- Pe is a standard measure of error rate of the system.
- It is on a bit-basis, so called as (Average) bit error rate
- For given n(t), good processing and opt-VT yield good Pe

- The BER is

- We assume that the source statistics are equally likely.

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise

- Two assumptions make things easy:
- 1. linear detection except for the threshold device
- 2. The AWGN channel white and Gaussian n(t) with zero mean and E{| n(t) |2}=σ02

- Notes:
- For baseband signaling, the processing circuits consisting of linear filter with some gain.
- For bandpass signaling, a superheterodyne circuit consisting of a mixer, IF stage, and product detector being also a linear circuit.
- If automatic gain control (AGC) or limiters or a nonlinear detector such as envelope detector is used, the results of this section will not be applicable.

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise

- For the case of a linear-processing receiver circuit with a binary signal plus noise at the input, the sampled output is:

- s0 is a constant that depends on the signal being sent

- Since the output noise n0 is a zero-mean Gaussian random variable, the total output sample r0 is a Gaussian random variable with a mean value of either s01 or s02, depending on whether a binary 1 or a binary 0 was sent. That is to say:

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise

- Then the two conditional PDFs are

- Using equally likely source statistics, the BER becomes:

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise

- thus

- To find the VT that minimizes Pe we need to solve dPe/dVT=0

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise and Matched-Filter Reception

- If the receiving filter is optimized the BER can be reduced. To minimize Pe, we need to maximize the argument of Q, thus we need to find the linear filter that maximizes:

- [sd(t0)]2 is the instantaneous power of the difference output signal at t=t0

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise and Matched-Filter Reception

- The linear filter that maximizes the instantaneous output signal power at the sampling t=t0 when compared with the average output noise power σ02=n02(t) is the matched filter. For the case of noise white noise at the receiver input, the matched filter needs to be matched to the difference signal sd(t)=s1(t)-s2(t). Thus the impulse response of the matched filter for binary signaling is:

- The output peak signal to average noise ratio that is obtained from the matched filter is:

7.1 Error Probabilities for Binary SignalingResults for Gaussian Noise and Matched-Filter Reception

- The performance is measured by

Pe=P(err|s1) P(s1) + P(err|s2) P(s2) => [P(err|s1) + P(err|s2) ]/2

- For AWGN channel, when linear detector used,

- More, optimal linear receiver or match filter C[s1(t0-t)-s2(t0-t)] can maximize the SNR of r0 and Pe even smaller

7.2 Performance of Baseband Binary Systems (Unipolar Signaling)

- Source data:

- Transmitted signal

- Input of receiver

- After sampling

- Data received

7.2 Performance of Baseband Binary Systems (Unipolar Signaling)

●Receive with (linear) LPF

● Detect with LPF of BHz to limit the noise and pass the signal

● Sampling at any time in the pulses, best at middle of them

● If gain=1, s01=A and s02=0,

Var of n0(t) = (2B) (N0/2) = N0B

● Decision with VT=A/2

For Rect-pulse, or for sin(x)/x pulse [B=1/(2T)]=A/2

7.2 Performance of Baseband Binary Systems (Unipolar Signaling)

Receive with (linear) Match filter

- Detect with filter: h(t) = s1(T-t)-s2(T-t) = A; In fact, this is a integrator! (Ed=A2T)
- Sampling at T (the end of pulses)
- s01=AT and s02=0;

Var of n0(t): do not care

- Decision with VT=AT/2
- Results

- Eb is the average bit-energe:
- No matter what pulse-shape, Pe is the same
- The best Pe, is always smaller than or equal to that of LPF-Pe.。

7.2 Performance of Baseband Binary Systems (Unipolar Signaling)

- For LPF, smaller BW introduces less noise and then good Pe. But, BW must be bigger enough to let the signal to pass.
- However, for MF the BW might be very large. For Example, the rect-pulse need rect-MF h(t). Its H(f) is a sinc-function and the absolute BW is infinitive! Then it would introduce much noise, why it could make the best Pe?

7.2 Performance of Baseband Binary Systems (Polar Signaling)

Bandwidth is BHz and the signal is low freq. in (0-B) Hz.

For Rect-pulses, B = 1/T Hz

7.2 Performance of Baseband Binary Systems (Polar Signaling)

- Source data:

- Transmitted signal

- Input of receiver

- After sampling

- Data received

7.2 Performance of Baseband Binary Systems (Polar Signaling)

●Receive with (linear) LPF

● Detect with LPF of BHz to limit the noise and pass the signal

● Sampling at any time in the pulses, best at middle of them

● If gain=1, s01=A and s02=-A,

Var of n0(t) = (2B) (N0/2) = N0B

● Decision with VT=0

For Rect-pulse, or for sin(x)/x pulse [B=1/(2T)]=A/2

7.2 Performance of Baseband Binary Systems (Polar Signaling)

Receive with (linear) Match filter

- Detect with filter: h(t) = [s1(T-t)-s2(T-t)]/2 = A; In fact, this is a integrator! (Ed=4A2T)
- Sampling at T (the end of pulses)
- s01=AT and s02=-AT;

Var of n0(t): do not care

- Decision with VT=0
- Results

- Eb is the average bit-energe:
- No matter what pulse-shape, Pe is the same
- The best Pe, is always smaller than or equal to that of LPF-Pe.。

7.2 Performance of Baseband Binary Systems (Bipolar Signaling)

- Read it youself with the following hints:
- 3-levels
- Calculate the conditional PDFs
- Get the best thresholds: VT1 and VT2, with the limitation: VT1=-VT2=VT
- Answer:
- What is thresholds: VT1 and VT2 ?
- What is Pe for normal LPF and match filter?
- What is the match filter h(t) ?

7.2 Performance of Baseband Binary Systems (Bipolar Signaling)

- Source data:

- Transmitted signal

- Input of receiver

- After sampling

- Data received

7.2 Performance of Baseband Binary Systems (Bipolar Signaling)

For the case of additive Gaussian noise, the BER is :

Where the optimum value of VT is

When σ0<<A, thus

7.3 Coherent Detection of Bandpass Binary Signals(On-Off Keying)

·The detection is mixer+H(f), which is linear processing

· Note the LO is 2cos(·) to keep gain of signal being 1

· LPF is to filter out the 2fc component produced by mixer

7.3 Coherent Detection of Bandpass Binary Signals (On-Off Keying)

- Source signal:

- Transmitted signal

- Input of receiver

- After sampling

- Data received

7.3 Coherent Detection of Bandpass Binary Signals (On-Off Keying)

Noise goes through mixer:

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7.3 Coherent Detection of Bandpass Binary Signals (On-Off Keying)

Receive with Mixer+LPF, (like baseband-unipolar case)

- Detect with LPF of BHz to filter off the high-freq, then unpolar-like pulses produced.
- Sampling at any time in the pulses, best at middle of them
- If gain=1, s01=A and s02=0;

Var of n0(t) = 2N0B

- Decision with VT=A/2
- Results

For Rect-pulse, or for sin(x)/x pulse [B=1/(2T)]=A/2

7.3 Coherent Detection of Bandpass Binary Signals (On-Off Keying)

Receive with (linear) Match filter

- Detect with filter

Ed=A2T/2

- Sampling at T (the end of pulses)
- s01=AT and s02=0; Var of n0(t): do not care
- Decision with VT=AT/2

- Eb is the average bit-energe:
- No matter what pulse-shape, Pe is the same
- The best Pe, is always smaller than or equal to that of LPF-Pe.。

7.3 Coherent Detection of Bandpass Binary Signals

Think about it :

- What is the similarity and difference between the OOK and Baseband unipolar signal ?

7.3 Coherent Detection of Bandpass Binary Signals (Binary-Phase-Shift Keying)

Read it youself with the following hints:

- Similar to that of Baseband Polar Signaling
- Note the bandpass (cos) characteristic
- Note the results

Answer:

- Pe for LPF and MF sturcture?
- What is the match filter h(t) ?
- What makes the difference on Pe between PSK(or Polar) and OOK(or Unipolar) , noise or Ed?

7.3 Coherent Detection of Bandpass Binary Signals (Binary-Phase-Shift Keying)

- Source data

- Transmitted signal

- Input of receiver

- After sampling

- Data received

7.3 Coherent Detection of Bandpass Binary Signals

Receive with Mixer+LPF, (like baseband-polar case)

- Detect with BPF of BHz, then polar-like pulses produced.
- Sampling at any time in the pulses, best at middle of them
- If gain=1, s01=A and s02=-A;

Var of n0(t) = 2N0B

- Decision with VT=A/2
- Results

For Rect-pulse, or for sin(x)/x pulse [B=1/(2T)]=A/2

7.3 Coherent Detection of Bandpass Binary Signals

Receive with (linear) Match filter

- Detect with filter

Ed=2A2T

- Sampling at T (the end of pulses)
- s01=AT and s02=-AT; Var of n0(t): do not care
- Decision with VT=AT/2

- Eb is the average bit-energe:
- No matter what pulse-shape, Pe is the same
- The best Pe, is always smaller than or equal to that of LPF-Pe.。

7.3 Coherent Detection of Bandpass Binary Signals (Frequency-Shift Keying)

7.3 Coherent Detection of Bandpass Binary Signals (Frequency-Shift Keying)

- Source data:

- Transmitted signal

- Input of receiver

- After sampling

- Data received

7.3 Coherent Detection of Bandpass Binary Signals (Frequency-Shift Keying)

Receive with Mixer+LPF, (like baseband-polar case)

- Detect with LPF of BHz to filter off the high-freqs, then polar-like pulses produced.
- Sampling at any time in the pulses, best at middle of them
- If gain=1, s01=+A and -s02=-A;

Var of n0(t) = 2x2N0B = 4N0B (doubled)

- Decision with VT=0
- Results
- For Rect-pulse, or for sin(x)/x pulse [B=1/(2T)]

7.3 Coherent Detection of Bandpass Binary Signals (Frequency-Shift Keying)

Receive with (linear) Match filter

- Detect with filter
- Sampling at T (the end of pulses)
- s01=AT and -s02=-AT;

Var of n0(t): do not care

- Decision with VT=0

7.3 Coherent Detection of Bandpass Binary Signals (Frequency-Shift Keying)

- Results
- Ed (max) = A2T, when f1-f2=n/(2T)=n(bit-rate/2); Then Pe is min as
- f1 and f2 is called orthogonal, when configured as above condition

Or, when f1-f2>>bit-rate, approximately orthogonal

- Eb is the average bit-energe:
- No matter what pulse-shape, Pe is the same
- This is best Pe, smaller than or equal to LPF-Pe

- Unlike OOK, FSK signal is constant-enveloped. Why Pe of FSK is not the same as PSK?

Summary

- When modulated signal is passed thru the AWGN channel, receiver with linear detector is often used
- The mixer+LPF provides a basic result:

Best Pe when sinc-pulse (smallest BW of LPF)

3dB worse when rect-pulse；even-worse if wider-BW pulses used

- The match filter=correlator, C[s1(T-t)-s2(T-t)] can minimize Pe, so best result
- Different modulations yield different Pe. With the same Eb/N0, means:

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