Derivation of expression for a Gaussian Filter with 3 dB bandwidth

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In GMSK modulation (used in GSM and DECT standard), a GMSK signal is generated by shaping the information bits in NRZ format through a Gaussian Filter. The filtered pulses are then frequency modulated to yield the GMSK signal. GMSK modulation is quite insensitive to non-linearities of power amplifier and is robust to fading effects. But it has a moderate spectral efficiency.

An expression for the Gaussian Filter with 3dB Bandwidth is derived here.

The requirements for a gaussian filter used for GMSK modulation in GSM/DECT standard  are as follows,

\(T = \text{ bit duration}\)

\(B =3\text{ dB Bandwidth of the filter}\)

\(BT =0.3 \text{ for GSM}\)

\(BT =0.5 \text{ for DECT}\)

Now the challenge is to design a Gaussian Filter fG(t) that satifies the 3dB bandwidth requirement i.e. in the frequency domain at some frequency f=B, the filter should posses -3dB gain ( in otherwords => half power point located at f=B)

The probability density function for a Gaussian Distribution with mean=0 and standard deviation=σ  is given by

\(f(t) = \frac{1}{\sqrt{2 \pi \sigma}} e^{ -\frac{t^{2}}{2 \sigma^{2}} }\)

The expression for the required Gaussian Filter can be obtained by choosing the variance of the above mentioned distribution so that the Fourier Transform of the above mentioned expression has a -3dB power gain at f=B.

The fourier transform of the above mentioned expression is

\(F[f(t)]=e^{-2 \sigma^{2} ( \pi f) }\)

Setting f=B,
\(e^{-2 \sigma^{2} ( \pi B) } = \frac{1}{\sqrt{2}} \Rightarrow \sigma = \frac{\sqrt{ln 2}}{2 \pi B}\)

\(\therefore f_{G}(t) = \sqrt{ \frac{2 \pi }{ ln 2}} B e^{ – \frac{2}{ln2}(\pi B t)^{2}} \)

See also :

[1] Correlative Coding – Modified Duobinary Signaling
[2] Correlative Coding – Duobinary signaling
[3] Nyquist and Shannon Theorem
[4] Correlative coding – Duobinary Signaling
[5] Square Root Raised Cosine Filter (Matched/split filter implementation)
[6] Introduction to Inter Symbol Interference

External Resources:

[1] The care and feeding of digital, pulse-shaping filters – By Ken Gentile
[2] Inter Symbol Interference and Root Raised Cosine Filtering – Complex2real

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