# SNR, BER, and Bandwidth Study Notes for GATE EC 2022 Exam

By BYJU'S Exam Prep

Updated on: September 25th, 2023

In this article, you will find the Study Notes on SNR, BER, and Bandwidth which will cover the topics such as Noise in Digital Communication, Pe for Non-coherent ASK and Non-coherent FSK, Comparison of Digital Modulation Schemes.

In this article, you will find the Study Notes on SNR, BER, and Bandwidth which will cover the topics such as Noise in Digital Communication, Pe for Non-coherent ASK and Non-coherent FSK, Comparison of Digital Modulation Schemes.

## Noise in Digital Communication

In digital communication for better SNR, matched filter is used whose impulse response h(t) is:

Consider a receiver model,

h(t) = Impulse response of LTI filter

x(t)=g(t) + ω(t)

T = Arbitrary observation interval

g(t) = Binary symbol 1 or 0

ω(t) = Sample function of white noise process of zero mean and power spectral density N0/2

Function of receiver is to detect the pulse signal g(t) from received signal x(t). The design of the filter is to be optimized to minimize the effects of noise.

As the filter is linear

∴            x(t) = g0(t) + n(t)

where go(t) and n(t) are produced signal and noise components of input x(t) respectively.

For proper noise free detection, signal component go(t) > > noise component n(t). This can be done by making the instantaneous power in output signal go(t) measured at time t = T as large as possible compared with average power of output noise n(t) i.e. peak pulse signal to noise ratio should be maximized.

Filter output due to noise ω(t) only is

For ⴄ to be maximum, we have to find H(f) for a given G(f).

By Schwarz inequality

Optimum of H(f) is given by

Hopt (f) = KG*(f)e–j2πfT

Taking inverse fourier transform

Probability of Error for Coherent ASK, PSK, FSK

In digital communication system, reliability is commonly expressed in terms of probability of bit error or bit error rate (BER) measured at the receiver output. Clearly, the smaller the BER, the more reliable the communication system is. The average bit error probability is denoted by Pe.

The probability of bit error for coherent ASK system is given by

Where Eb is the bit energy, N0 is the noise power density, and γb is the bit energy to noise density ratio.

·           In PSK

If distance between two manage points is decreased, then probability of error will increase.

·           For FSK (coherent) probability of error is

For coherent binary FSK signal, we define the bit error probability as

Where Eb is the bit energy, N0 is the noise power density, γb is the bit energy to noise density ratio.

Note:

For larger value of z, the Q(z) function can be approximated as

## Pe for Non-coherent ASK and Non-coherent FSK

·           Probability of error for non-coherent ASK is

if

·           Probability of error for non-coherent FSK is

Probability of error for differential PSK is

• In case of FSK f1 and f2 are chosen such that f1 = mfs and f2 = kfs where m, k are integers.
• Bandwidth efficiency

## Comparison of Digital Modulation Schemes

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