2. Wendt 2
Abstract:
The purpose of this lab was to create a circuit capable of discerning a single tone message
within a FM wave. There are many ways of extracting the data within a FM wave SFM(t)). For
this experiment we used the method of differentiating, enveloping, and processing the resultant
signal through a bandpass to extract the FM message signal (m(t)). This report serves to explain
our method as well as provide the theory behind the calculations. Using PSPICE, we were able to
simulate our desired circuit based on ideal circuit operation. We were able to extract a clean
message signal at the maximum message frequency (fm) of 1000 Hz. However, the system failed
for the lowest fm within our range, 100 Hz. I believe this is due to incorrect usage of the high-
pass filter as we used a natural frequency of 1000 Hz in determining our circuit element values
which may have caused a clipping effect on the desired message signal. I think reducing the
natural frequency of the high-pass filter by increasing the capacitor values could rectify the
problem. This report includes an introduction, an overview of the theory, a list of references for
borrowed concepts, and an appendix with the code file of our PSPICE model as well as the
PSPICE model image.
3. Wendt 3
Introduction:
A frequency modulated (FM) wave contains data within the variation of the waveβs frequency.
The element of the FM wave responsible for the variable frequency is called the message signal.
In the case of this lab we used a sinusoidal m(t) to drive the sinusoidal SFM(t). There are many
ways to extract the message signal. In this experiment we used the method of differentiation. The
message signal is included in the radial frequency component of the transient sinusoidal FM
wave. As such, we can use differentiation to incorporate the frequency component if the sinusoid
into the wave amplitude.
(1)
ππ πΉπ(π‘)
ππ‘
=
π
ππ‘
(π΄ πcos [2πππ π‘ + 2ππΎπ β« π(π₯)ππ₯
π‘
0
])
= π΄ π(2ππΎπ π(π‘) + πΏ)cos(2πππ π‘ + 2ππΎπ β« π(π₯)ππ₯
π‘
0
+
π
2
)
Equation (1) transforms the purely FM signal into both AM (amplitude modulated) and
FM. We can then used an envelope detector to trace the amplitude of the differentiated signal.
We constructed a low-pass filter to eliminate the component responses due to frequencies higher
than that of the desired message signal (like that from the carrier frequency. Finally, the signal is
processed through a high-pass filter to eliminate any DC components that may have been added
throughout the signal processing. A diagram of the proposed circuit is included below:
We implemented and LC tank in front of our active differentiator circuit in order to curb
the effects of frequency on our transient signal. The envelope is simply a diode in series with a
capacitor and resistor in parallel to ground. The output is the voltage output of the diode which,
loaded by the capacitor/resistor combo, fluctuates around the amplitude of the positive output
from the differentiator. We used active 2nd
order Butterworth low-pass and high-pass filters to
refine the message signal. By running two low-pass filters in series we were able to accomplish a
significantly smoother signal. The high-pass filter simply served to adjust the offset of the signal.
The circuit elements in each component of our FM receiver were precisely chosen to achieve a
particular set of buffering criteria.
4. Wendt 4
Theory:
This section serves as an overview of the circuit theory as well as a guide through the
mathematical derivations used to obtain our circuit element values. The first series of equations
shows the evolution of the signal as it progressed through the receiver.
Equation (1) shown above shows the first transformation of the signal. After being
processed through the differentiator, SFM(t) becomes S1(t):
π΄ π(2ππΎπ π(π‘) + πΏ)cos(2πππ π‘ + 2ππΎπ β« π(π₯)ππ₯
π‘
0
+
π
2
As explained in the introduction, the purpose of the differentiation is to extract the message
signal from the frequency component of the sinusoid, multiplying it and the constant carrier
frequency (π c) into the amplitude of the sinusoid. The addition of
π
2
serves correct the sinusoidal
portion of differentiation. πΏ is related to the maximum overall frequency deviation by the
equality:
πΏ > 2πβπ
This equality ensures that the envelope is always greater than 0. βπ can be found by:
βπ = π½ππ
Where π½ is the FM modulation index (chosen to be π½ = 4) and ππ is the message signal
frequency.
Using a diode, the envelope only passes positive voltages causing an intermittent signal.
This charges the capacitor while passing and allows the capacitor to discharge when the signal is
stopped. Depending on the values chosen for R and C, the resulting voltage signal will fluctuate
around the amplitude of the passed FM signal. Values for R and C were chosen with respect to
the various frequencies of the signal and the time constant, Ο = 1 / RC. To ensure that the
capacitor does not discharge faster than the period of the message signal but slower than the
period of the carrier signal. The values were determined from the following inequality:
1
ππ
βͺ π πΆ βͺ
1
π π
5. Wendt 5
The Butterworth Low-Pass circuit values were chosen from the s domain transfer
function of the following circuit:
The transfer function is as follows:
π»(π ) =
π2
π 2 + 2ππ π + π2
=
(
1
π 2 πΆ1 πΆ2
)
π 2 +
2
π πΆ1
π + (
1
π 2 πΆ1 πΆ2
)
From this we derived:
π2
= (
1
π 2 πΆ1 πΆ2
)
And:
β2
2
π =
1
π πΆ1
Combining yielded:
πΆ1 = 2πΆ2
We chose C values based on that relation and then chose R using π = 2Ο1000.
6. Wendt 6
The calculations and circuit for the high-pass Butterworth were very similar. The circuit
is as follows:
The transfer function is as follows:
π»(π ) =
π 2
π 2 + 2ππ π + π2
=
π 2
π 2 +
2
πΆπ 2
π + (
1
πΆ2 π 1 π 2
)
π2
= (
1
πΆ2 π 1 π 2
)
β2
2
π =
1
πΆπ 2
Yielding:
π 2 = 2π 1
We chose R values based on that relation and C using π = 2Ο1000.
7. Wendt 7
The simulation yielded desirable results. The figure below shows SFM(t) (blue),
differentiated signal (green), and the output of the encoder (red).
Next is the output from the Butterworth low-pass filters:
8. Wendt 8
Finally, we have the output of the Butterworth high-pass i.e. the message signal, m(t):
To get these results we set the carrier frequency to 10000 Hz, Ξ² = 4, and message
frequency = 1000 Hz. When we applied a 100 Hz message frequency we obtained the following
skewed message signal:
9. Wendt 9
References:
Much of the information used in this report was obtained from notes taken from Dr. Ben
Belzer found at: http://eecs.wsu.edu/~ee352/labassigns/project/project_notes.pdf
