19 Feb 2019

Hilbert Transform and Inverse Hilbert Transform (Definition, Formula, Basics and Properties of Hilbert Transform)

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).

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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.

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Carson's Rule for Bandwidth Calculation in Frequency Modulation (Narrowband and Wideband FM) - Carson's Formula

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.

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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


Derivation of Carson's Rule for Narrowband FM and Wideband FM


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.

Read More-

Go To HOME Page
   
FREQUENCY SPECTRUM OF AMPLITUDE MODULATION (WAVEFORMS AND EQUATIONS DERIVATION)

AMPLITUDE MODULATION (TIME DOMAIN EQUATIONS AND WAVEFORMS)

ADVANTAGES AND DISADVANTAGES OF DIGITAL COMMUNICATION SYSTEM

ADVANTAGES OF OPTICAL FIBER COMMUNICATION

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)

AMPLITUDE MODULATION Vs FREQUENCY MODULATION (ADVANTAGES AND DISADVANTAGES)

PULSE CODE MODULATION (PCM) [ADVANTAGES AND DISADVANTAGES]

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)

Quadrature Amplitude Modulation (QAM)/ QAM Transmitter and QAM Receiver Block Diagram

Single-Mode Optical Fiber Advantages

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

Microwaves Properties and Advantages (Benefits)

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)

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