## What is Hilbert Transform?

If x(t) is a signal and we represent its hilbert transform by xh(t), then xh(t) is obtained by providing '/2' phase shift to every frequency component present in x(t).

Watch the Complete Video Here

Now let's see formula to calculate hilbert transform and inverse hilbert transform.

## Hilbert Transform Formula

With the help of following formula we can easily calculate the hilbert transform

## Inverse Hilbert Transform Formula

The formula provided here can be used to find the inverse hilbert transform

Now let's discuss properties of hilbert transform

## Properties of Hilbert Transform

1. A signal x(t) and its hilbert transform xh(t) have the same energy density spectrum.

2. A signal x(t) and it's a hilbert transform xh(t) have the same autocorrelation function

3. A signal x(t) and its hilbert transform xh(t) are mutually orthogonal. We can write it mathematically as-

4. If xh(t) is a hilbert transform of x(t) then the hilbert transform of xh(t) is -x(t).

It means -

if H[x(t)] = xh(t)

then H[xh(t)] = -x(t)

Here 'H' denotes the Hilbert transform.

FREQUENCY SPECTRUM OF AMPLITUDE MODULATION (WAVEFORMS AND EQUATIONS DERIVATION)

AMPLITUDE MODULATION (TIME DOMAIN EQUATIONS AND WAVEFORMS)

STEP INDEX OPTICAL FIBER (MULTIMODE AND SINGLE MODE STEP INDEX FIBERS)

PULSE MODULATION TECHNIQUES (PAM, PWM, PPM, PCM)

OPTICAL FIBER: STRUCTURE AND WORKING PRINCIPLE

PULSE AMPLITUDE MODULATION (PAM)

COMPARISON OF PAM, PWM, PPM MODULATION TECHNIQUES

PULSE WIDTH MODULATION (PWM)

CONTINUOUS TIME AND DISCRETE TIME SIGNALS (C.T. AND D.T. SIGNALS)

NEED AND BENEFITS OF MODULATION

PULSE POSITION MODULATION (PPM)

OPTICAL FIBERS IN COMMUNICATION: COVERS ALL IMPORTANT POINTS

OPTICAL FIBER SOURCES (DESIRABLE PROPERTIES)

SAMPLING THEOREM AND RECONSTRUCTION (SAMPLING AND QUANTIZATION)

SUPERPOSITION THEOREM (BASICS, SOLVED PROBLEMS, APPLICATIONS AND LIMITATIONS)

Digital Modulation Techniques (ASK, FSK, PSK, BPSK)/ Amplitude, Frequency and Phase Shift Keying

Conventional AM Vs DSB-SC Vs SSB-SC Vs VSB (Comparison of AM Systems)

What are Microwaves and their Applications (Uses) in various fields

Basic Structure of Bipolar Junction Transistor (BJT) - BJT Transistor - Working and Properties

Polar Plots of Transfer Functions in Control Systems (How to Draw Nyquist Plot Examples)

Generation of Binary Phase Shift Keying (BPSK Generation) - Block Diagram of Binary Phase Shift Keying (BPSK)

Low Level and High Level Modulation Block Diagram (AM Transmitter Block Diagram)

Block Diagram of CRO (Cathode Ray Oscilloscope), Components of CRO and CRT with Structure and Working

Slope Overload Distortion and Granular (Idle Noise), Quantization Noise in Delta Modulation

Frequency Translation/Frequency Mixing/Frequency Conversion/Heterodyning (Basic Concepts and Need)

Quadrature Phase Shift Keying Modulation (QPSK) Basics, Waveform and Benefits

Pulse Code Modulation (PCM) Vs Differential Pulse Code Modulation (DPCM)

## What is Carson's Rule?

Carson's formula is used to calculate the bandwidth (BW) of a single tone wideband FM.
According to carson's rule, the FM bandwidth is given as, twice the sum of frequency deviation and the highest modulating frequency.
But it should be noted here that this rule is just an approximation.

Watch the Complete Video Here

So Carson's rule can be written mathematically as-

BW = 2(∆w + wm)

But mf = ∆w/wm

Therefore

BW = 2(mfwmwm)

= 2wm (mf + 1)

Now we have two special cases for the carson's rule -

1 - If ∆w << wm and

2- ∆w >> wm

Case 1- If ∆w << wm

Since mf = ∆w/wm

If ∆w << wm

=> mf << 1

=> It is the case for narrowband FM

Since the bandwidth by the carson's rule is given
as-

BW = 2(∆w + wm)

BW = 2(mfwmwm)

= 2wm (mf + 1)

Therefore for m<< 1

BW = 2wm

Note here that this is equivalent to Amplitude Modulation (AM)

Case 2- ∆w >> wm

Since mf = ∆w/wm

Therefore if ∆w >> wm

=> m>> 1   as is the case for wideband FM
Then, since by Carson's rule

BW = 2wm (mf + 1)

Therefore for mf  >> 1

BW = 2wmmf

But wmmf = ∆w

Therefore

BW = 2∆w

Note- For large values of m this BW relationship can be considered accurate for all practical purposes.

FREQUENCY SPECTRUM OF AMPLITUDE MODULATION (WAVEFORMS AND EQUATIONS DERIVATION)

AMPLITUDE MODULATION (TIME DOMAIN EQUATIONS AND WAVEFORMS)

STEP INDEX OPTICAL FIBER (MULTIMODE AND SINGLE MODE STEP INDEX FIBERS)

PULSE MODULATION TECHNIQUES (PAM, PWM, PPM, PCM)

OPTICAL FIBER: STRUCTURE AND WORKING PRINCIPLE

PULSE AMPLITUDE MODULATION (PAM)

COMPARISON OF PAM, PWM, PPM MODULATION TECHNIQUES

PULSE WIDTH MODULATION (PWM)

CONTINUOUS TIME AND DISCRETE TIME SIGNALS (C.T. AND D.T. SIGNALS)

NEED AND BENEFITS OF MODULATION

PULSE POSITION MODULATION (PPM)

OPTICAL FIBERS IN COMMUNICATION: COVERS ALL IMPORTANT POINTS

OPTICAL FIBER SOURCES (DESIRABLE PROPERTIES)

SAMPLING THEOREM AND RECONSTRUCTION (SAMPLING AND QUANTIZATION)

SUPERPOSITION THEOREM (BASICS, SOLVED PROBLEMS, APPLICATIONS AND LIMITATIONS)

Digital Modulation Techniques (ASK, FSK, PSK, BPSK)/ Amplitude, Frequency and Phase Shift Keying

Conventional AM Vs DSB-SC Vs SSB-SC Vs VSB (Comparison of AM Systems)

What are Microwaves and their Applications (Uses) in various fields

Basic Structure of Bipolar Junction Transistor (BJT) - BJT Transistor - Working and Properties

Polar Plots of Transfer Functions in Control Systems (How to Draw Nyquist Plot Examples)

Generation of Binary Phase Shift Keying (BPSK Generation) - Block Diagram of Binary Phase Shift Keying (BPSK)

Low Level and High Level Modulation Block Diagram (AM Transmitter Block Diagram)

Block Diagram of CRO (Cathode Ray Oscilloscope), Components of CRO and CRT with Structure and Working

Slope Overload Distortion and Granular (Idle Noise), Quantization Noise in Delta Modulation

Frequency Translation/Frequency Mixing/Frequency Conversion/Heterodyning (Basic Concepts and Need)

Quadrature Phase Shift Keying Modulation (QPSK) Basics, Waveform and Benefits

Pulse Code Modulation (PCM) Vs Differential Pulse Code Modulation (DPCM)