# ANALYSIS OF LINEAR FM SIGNAL INFLUENCE ON NARROW BAND LINEAR CIRCUIT

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```					           ANALYSIS OF LINEAR FM SIGNAL INFLUENCE ON NARROW BAND
LINEAR CIRCUIT

E. Bakin, A. R. Zhezherin

St. Petersburg State University of Aerospace Instrumentation
67 Bolshaya Morskaya str., 190000, St. Petersburg, RUSSIA,

In modern security alarm-systems narrow band signals are used to transfer data from object
to central supervision station. For example different kinds of amplitude-impulse modulation or
frequency modulation with small modulation index are used. However, nowadays, the broadband
systems with the channel (subscriber) code division principles get more and more popularity. For
example principles of code channel division CDMA is widely used in cellular phone systems and
a few security systems (for example Spread Net system by S&K). Subscriber code division
principles introduction goes simultaneously with big number already created narrow-band
systems using. So, researching the narrow-band system reaction on wide-band signal is a
problem of great interest.
In this work, current generator, producing linear-FM radio-impulses, was used as a model
of wide-band signal. High-quality parallel oscillatory circuit was used as a model of narrow-band
system. Researching model scheme is shown on fig. 1.

Fig.1 Narrow-band model scheme.

The case, when circuit resonant frequency was equal to linear-FM central frequency, was
considered. In this case it is expediently to use frequency-selective circuit low-frequency (LF)
equivalent for output voltage envelope shape analysis. LF-equivalent is an imaginary system,
whose gain can be found by moving the real narrow-band system gain from resonant frequency
vicinity to zero frequency vicinity:
K LF ( jΩ) = K [ j (ωR + Ω)] ,              [1]

where ωR is a resonant frequency of real narrowband system.
Input resistance is taken as frequency characteristic.
R0 r
Z ( jω ) =
2 jQ(ω − ωR ) ,        [2]
1+
ωR
L
where   R0 r =          is a circuit resonant resistance, L – Inductivity, C – capacity, r – circuit
C ⋅r
active resistance.
LF-equivalent characteristics can be found, making the variable changing:
ω = ωR + Ω
R0 e      R0 r
Z LF ( jΩ) =                         =
2 jQΩ (1 + jτ C Ω )              ,   [3]
1+
ωR
where τC = 2Q / ωR – time constant of damping, Ω - current frequency.
LF-equivalent impulse characteristic can be found by using reversed Fourier transform for
Z LF ( jΩ) :
∞

∫∞ (1 + jτ C Ω) exp[ jΩt ]⋅ dΩ .
1                R0 r
hLF (t ) =
2π        −
[4]

So, oscillated contour impulse characteristics:
2 ⋅ R 0r              ⎡-t ⎤
h LF (t) =                   ⋅ exp ⎢ ⎥ ⋅1(t) .
τC                  ⎣τ C ⎦
[5]

It’s clear that impulse characteristics of real contour is radio signal
h(t) = h LF (t ) ⋅ cos(ωRt ) ⋅1(t ) , and hLF is envelope of this signal.
For calculating radio signal voltage envelope on oscillator contour (on system output),
when the currency impulse is on input, we use method of complex envelopes.
Output signal can be written in such a way:
t

U out (t ) = ∫ I IN (τ ) ⋅ h LF (t - τ ) dτ .                                [6]
−∞
For this task, we can write:
t                    μτ 2
j
U out (t ) =    ∫I       0
e        2
⋅ h LF (t - τ ) dτ ,             [7]
−ti
2

2π ⋅ Δf
where   μ=                , Δf – frequency deviation within impulse, ti – impulse duration,
ti
⎡ μ ⋅t2 ⎤
I in (t ) = I 0 exp⎢ j     ⎥ – input currency linear-FM impulse complex envelope.
⎣ 2 ⎦
Plot of frequency as a function of time is shown on fig. 2.

fig.2 Plot of frequency as a function of time

Contour parameters L and C where chosen, proceeding from condition that fr = f0 . Contour
band pass (0.707 level) ΔfBP was regulated by changing active resistance r, Δf , frequency
deviation was changed in range beginning from 0.1 to 10 MHz. System band pass was changes
in range beginning from 10 to 100 kHz.
A few calculations with different values contour band pass, marked as BP, and frequency
deviation within input current impulse were made. Results of calculation are given on fig. 3-5.
On these plots, numbers 1-5 mean different values of frequency modulation.
For comparison, the output voltage envelopes when the rectangle current radio impulse
without deviation (test-impulse) is at input are shown on the plots. f0=fR and spectrum active
width of this test-impulse is less then ΔfBP. Length of this test-impulse is equal to length of
linear-FM impulses (in this case energies of both impulses are equal). With these signal
parameters, test-signal voltage envelope reaches its maximum value. This value was the initial
one and all envelopes were considered relatively to it.
fig.3 Plot of output voltage envelope. BP = 10 KHz

fig.4   Plot of output voltage envelope. BP = 40 KHz
fig.5 Plot of output voltage envelope. BP = 100 KHz

Conclusions
Maximum value of narrow band system output voltage decreases much in comparison with
radio signal without frequency modulation as a result of growing of frequency changing speed
within input current impulse. When the pass band is 10 KHz (that is equal to real narrow band
security systems pass band) and the frequency deviation is 10 MHz, attenuation reaches the
value of 20 dB.
As a result of band pass broadening, the signal attenuates significantly weaker. This may be
the cause of the interference to the security system operation.

References.
1. Baskakov S. I., “Radiotechnical circuits and signals”, Moscow, “Vischaya Shkola”, 2003.
448 pages.