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__FREQUENCY RESPONSE / NORMALISATION AND FILTER
CIRCUITS__

**Frequency response**

It is defined as the *magnitude* (*gain*),
and *phase* differences between the input and the output sinusoids. To plot
the frequency response, a vector of frequencies is created first (varying
between zero or "DC" and infinity), and compute the value of the transfer
function at those frequencies. If **G(S)** is the transfer function of a
system and **w** is the frequency vector, the **G(jw)** vs **w** is
then plotted. Since **G(jw)** is a complex number, both magnitude and phase
responses can be plotted (bode plots).

**Bode Plots
**are the *magnitude (gain) *and *phase *response curves as
functions of **log(w)** (frequency). Every
system has a magnitude and phase response where they describe the system.
Sketching Bode plots can be simplified because they can be approximated as a
sequence of straight lines. Straight-line approximation simplify the evaluation
of the magnitude and frequency response. Usually the gain in
decibels, abbreviated **dB**, and the phase are plotted linearly along the
y-axis on graph paper that has several cycles of a logarithmic scale on the
x-axis. Each cycle represents a factor of 10 in frequency.

**Cut-off frequency** is the final point at
which the filter response drops 3dB or to 0.707 of its peak value.

**Decade** is a logarithmic way of measuring
the gain or loss. Decibels are defined as **20log**_{10 }of a
voltage ratio **(V**_{o}/V_{i}), which is the division of the
output voltage with the input voltage.

The **gain margin** is found by using the phase
plot to find the frequency **W**_{GM}, where
the phase angle is **180°**. This is shown in *figure G* below . By
observing the magnitude plot at this frequency, the Gain margin **G**_{M}
can be determined, which is the gain required to raise the magnitude curve to**
0 dB**, shown in yellow colour.

The **phase margin** can be found by using the
magnitude curve to find the frequency **W**_{PM},
where the gain is **0 dB**. By then looking on the phase curve at that
frequency, the Phase margin **P**_{M}, is the difference between the*
phase value and 180°*, shown in fuchsia colour. This way, every system
characterised by bode plots as gain and phase plots, it's gain and phase margins
can be found easily.

*figure G : gain & phase
margins*