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Effect of Noise in Amplitude Modulation (AM) Explained

 

Effect of Noise in Amplitude Modulation (AM) 

Understanding the effect of noise in amplitude modulation (AM) is one of the most important topics in Analog Communication Systems. 

This article explains the complete derivation step by step, including:

  • Transmitted AM signal
  • Received signal equation
  • Why noise is written in envelope-phase form
  • Difference between r(t) and y(t)
  • Envelope detector output
  • Threshold effect

Step 1: Transmitted AM Signal

The transmitted amplitude modulated (AM) signal is

s(t)=Ac[1+kam(t)]cos(2Ï€fct)

where

  • Ac = Carrier amplitude
  • m(t) = Message signal
  • ka = Amplitude sensitivity
  • fc = Carrier frequency

Step 2: Noise is Added by the Channel

Every communication channel introduces random noise. Let the channel noise be

n(t)

The received signal becomes

r(t)=s(t)+n(t)

This is the most fundamental equation in communication systems.

Received Signal = Transmitted Signal + Channel Noise

Step 3: Represent Noise Around the Carrier Frequency

Instead of writing noise directly as n(t), narrowband noise is represented as

n(t)=nI(t)cos(2Ï€fct)-nQ(t)sin(2Ï€fct)

where

  • nI(t) = In-phase noise
  • nQ(t) = Quadrature noise

These are low-frequency random signals.

Step 4: Convert Noise into Envelope-Phase Form

The same narrowband noise can also be represented as

n(t)=r(t)cos[2πfct+ψ(t)]

where

  • r(t) = Noise envelope (noise amplitude)
  • ψ(t) = Noise phase
Important:

The symbol r(t) used here represents the noise envelope, not the received signal. 

Step 5: Add Signal and Noise

The received waveform can now be written as

y(t)=s(t)+n(t)

Substituting the signal and noise expressions gives

y(t)=Ac[1+kam(t)]cosωct +r(t)cos(ωct+ψ(t))

This is the complete received signal.

Step 6: Expand the Noise Component

Using the trigonometric identity

cos(A+B)=cosAcosB−sinAsinB

we obtain

y(t)= [Ac(1+kam)+rcosψ]cosωct -rsinψsinωct

The received signal now consists of two noise components.

  • Noise multiplying cos(ωct)
  • Noise multiplying sin(ωct)

Step 7: Why is One Noise Component Ignored?

When the carrier amplitude is much larger than the noise,

Ac >> r(t)

the envelope detector mainly responds to amplitude variations.

The term

r(t)cosψ

directly changes the envelope.

The term

r(t)sinψ

mainly changes the phase of the carrier.

Since an envelope detector is insensitive to phase changes, the quadrature component is neglected.

Step 8: Approximate Received Signal

The received signal becomes

y(t)≈ [Ac(1+kam)+r(t)cosψ(t)]cosωct

Notice that the noise simply adds to the envelope.

Step 9: Envelope Detector Output

The envelope detector removes the carrier component.

The detected envelope is

Ac(1+kam)+r(t)cosψ(t)

Removing the DC carrier component gives

Ackam(t)+r(t)cosψ(t)

Therefore

  • Desired Signal: Ackam(t)
  • Noise: r(t)cosψ(t)

Step 10: Threshold Effect in AM

Everything discussed so far assumes that the carrier is much stronger than the noise.

Ac >> r(t)

If the carrier becomes weak or the noise becomes very large,

r(t)≈Ac

then

  • The quadrature noise can no longer be ignored.
  • The envelope becomes highly distorted.
  • The detector follows random peaks instead of the actual envelope.
  • The output SNR suddenly drops.

This sudden degradation in performance is called the Threshold Effect.

Overall Flow of Noise in AM

Transmitter
     │
     ▼
AM Signal

s(t)

     │
     ▼
Channel Adds Noise

     │
     ▼
Received Signal

y(t)=s(t)+n(t)

     │
     ▼
Represent Noise

n(t)=r(t)cos(ωct+ψ)

     │
     ▼
Expand Using Trigonometry

     │
     ▼
Ignore Quadrature Noise
(Strong Carrier)

     │
     ▼
Envelope

Ac(1+kam)+r cosψ

     │
     ▼
Envelope Detector

     │
     ▼
Output

Ac kam(t)+r cosψ

     │
     ▼
Output SNR

     │
     ▼
If Carrier Becomes Weak

     │
     ▼
Threshold Effect
  • The received signal is always the transmitted signal plus channel noise.
  • Narrowband noise is represented in envelope-phase form for easier analysis.
  • The envelope detector is mainly affected by the in-phase noise component.
  • The quadrature component is neglected only when the carrier is much stronger than the noise.
  • The detected output consists of the desired message plus the in-phase noise component.
  • When the carrier becomes weak, the envelope detector fails, resulting in the Threshold Effect.


What Does a Strong Carrier Mean in Amplitude Modulation (AM)?

One of the most common misconceptions in AM noise analysis is the phrase strong carrier. It does not necessarily mean that the carrier gains more energy during transmission.

Instead, a strong carrier means that the carrier amplitude is much larger than the noise amplitude at the receiver.

Case 1: High Carrier Power

Suppose the transmitter sends a carrier with the following power:

  • Carrier Power = 100 W
  • Noise Power = 1 W
Carrier-to-Noise Ratio = 100 / 1 = 100

Since the carrier power is much greater than the noise power, the carrier is considered strong. The received waveform is dominated by the carrier, making it easier for the envelope detector to recover the message signal.

Case 2: Carrier Weakens During Transmission

As the signal propagates through the communication channel, attenuation reduces the carrier power.

  • Carrier Power = 2 W
  • Noise Power = 1 W
Carrier-to-Noise Ratio = 2 / 1 = 2

Now the carrier is only slightly stronger than the noise. The envelope detector finds it much more difficult to distinguish the message from the noise.

Mathematical Meaning of a Strong Carrier

The transmitted AM signal is

s(t)=Ac(1+kam(t))cos(ωct)

The narrowband noise is represented as

n(t)=r(t)cos(ωct+ψ)

where

  • Ac = Carrier amplitude
  • r(t) = Noise envelope (noise amplitude)

A strong carrier simply means

Ac >> r(t)

In other words, the carrier amplitude is much greater than the instantaneous noise amplitude.

Important:

A strong carrier does not necessarily mean that the carrier itself has become stronger. It simply means that the carrier is much stronger relative to the noise.

Why Can We Ignore the Phase Noise Component?

After expanding the received signal, we obtain

y(t)= [Ac(1+kam)+r cosψ]cosωct -r sinψ sinωct

The two noise terms have different effects:

  • r cosψ changes the amplitude (envelope).
  • r sinψ mainly changes the phase of the carrier.

When

Ac >> r

the phase variation is extremely small compared with the carrier amplitude. Since an envelope detector responds primarily to amplitude variations, the phase noise has very little effect and can be neglected.

When Does the Threshold Effect Occur?

Example 1: Strong Carrier

  • Carrier Amplitude = 10 V
  • Noise Amplitude = 0.2 V
Ac >> r

The envelope detector accurately follows the envelope, resulting in good signal recovery.

Example 2: Weak Carrier

  • Carrier Amplitude = 0.5 V
  • Noise Amplitude = 0.4 V
Ac ≈ r

Now the noise is almost as large as the carrier. The detector can no longer distinguish the true envelope from the noise, causing severe distortion in the recovered signal.

This sudden degradation in receiver performance is known as the Threshold Effect.

Visual Comparison

Strong Carrier

Carrier Amplitude : ||||||||||||||||||||
Noise Amplitude   : ||

Result:
Carrier dominates.
Envelope detector works correctly.


Weak Carrier

Carrier Amplitude : ||||
Noise Amplitude   : |||

Result:
Noise competes with carrier.
Envelope becomes distorted.
Threshold Effect occurs.

Summary

  • A strong carrier means the carrier amplitude is much larger than the noise amplitude.
  • It does not necessarily mean that the carrier has gained more energy.
  • The condition for a strong carrier is Ac >> r(t).
  • Under this condition, phase noise is negligible and only amplitude noise affects the envelope detector.
  • When the carrier becomes comparable to the noise, the envelope detector fails, leading to the threshold effect.
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