In reality, multiple radio stations operate in the same metro area, or market. When you tune in a signal at 750 kHz, another signal may be at 760 kHz. To find out whether the adjacent signal impacts the simple receiver design, assume that the interference is a single tone, Aicos(2πfit). The received signal is now of the form
![image0.jpg](https://www.dummies.com/wp-content/uploads/379617.image0.jpg)
with fi assumed to lie just outside the +/– 5-kHz channel bandwidth centered on fc.
Using a generalized version of the phasor addition formula, you can show that the received envelope with single-tone interference is as follows:
![image1.jpg](https://www.dummies.com/wp-content/uploads/379618.image1.jpg)
Note that
![image2.jpg](https://www.dummies.com/wp-content/uploads/379619.image2.jpg)
The envelope detector recovers envelope R(t). To find R(t) for a new signal model, use the phasor additional formula, which can be shown to hold for time-varying amplitudes and phases of the constituent terms. The enhanced formula states that
![image3.jpg](https://www.dummies.com/wp-content/uploads/379620.image3.jpg)
where
![image4.jpg](https://www.dummies.com/wp-content/uploads/379621.image4.jpg)
The key to this formula working is the common f0 found in each cosine term.
For AM plus single-tone interference, you can make the formula work by adding and subtracting fc in the interference term:
![image5.jpg](https://www.dummies.com/wp-content/uploads/379622.image5.jpg)
In the formula A1(t) = Ac[1 + am(t)],
![image6.jpg](https://www.dummies.com/wp-content/uploads/379623.image6.jpg)
Now, calculate
![image7.jpg](https://www.dummies.com/wp-content/uploads/379624.image7.jpg)
Add these complex numbers in rectangular form and then find the magnitude:
![image8.jpg](https://www.dummies.com/wp-content/uploads/379625.image8.jpg)
The last line follows from
![image9.jpg](https://www.dummies.com/wp-content/uploads/379626.image9.jpg)
Because
![image10.jpg](https://www.dummies.com/wp-content/uploads/379627.image10.jpg)
you can combine the terms and drop the absolute value. As a check, if Ai = 0, that is, no interference, the result for R(t) reduces to
![image11.jpg](https://www.dummies.com/wp-content/uploads/379628.image11.jpg)
The envelope detector is relatively easy to implement in hardware, but it’s a little difficult to analyze. You can explore the model for R(t) to get a feel for what’s going on. For starters, the input/output relationship is nonlinear, as evidenced by the squares and square root operations. Even with Ai = 0, R(t) contains an absolute value. At this point, assume m(t) = cos(2πfmt) as simple test case.
The Python function env_plot(t,Ac,Am,fm,Ai,fi) allows R(t) to be plotted as well as its spectrum. The spectrum PR(f) is a result of using PyLab’s psd() function.
In [<b>346</b>]: def env_plot(t,Ac,Am,fm,Ai,fi): ...: R = sqrt((Ac+Am*cos(2*pi*fm*t) +Ai*cos(2*pi*dfi*t))**2 +(Ai*sin(2*pi*dfi*t))**2) ..: return R In [<b>347</b>]: t = arange(0,20,1/500.) # T=20ms, fs=500 kHz
Exercise the function by using a time vector running over 20 ms at an effective sampling rate of 500 ksps and then plot time-domain and frequency-domain results side by side (see the 3 x 2 subplot array in the figure).
Set
![image12.jpg](https://www.dummies.com/wp-content/uploads/379629.image12.jpg)
Also set Am = 0.5, which is equivalent to setting a = 0.5 (50 percent modulation depth). The value of Ai steps over 0, 0.1, and 1.0. The 2-kHz message is within the 5-kHz message bandwidth requirement and
![image13.jpg](https://www.dummies.com/wp-content/uploads/379630.image13.jpg)
kHz for the interference places it in the adjacent channel (5 kHz is the crossover frequency).
Here are the primary IPython command line entries:
In [<b>447</b>]: R = env_plot(t,1,.5,2,0,7) In [<b>449</b>]: plot(t,R) In [<b>454</b>]: psd(R,2**13,500); In [<b>457</b>]: R = env_plot(t,1,.5,2,0.1,7) In [<b>459</b>]: plot(t,R) In [<b>464</b>]: psd(R,2**13,500); In [<b>467</b>]: R = env_plot(t,1,.5,2,1,7) In [<b>469</b>]: plot(t,R) In [<b>475</b>]: psd(R,2**13,500);
![[Credit: Illustration by Mark Wickert, PhD]](https://www.dummies.com/wp-content/uploads/379631.image14.jpg)
The only way to eliminate or reduce the interference is with a BPF in front of the envelope detector. The superheterodyne option is a good choice here, because the BPF doesn’t need to be tunable. At first, you may be disturbed that an out-of-band signal can produce in-band interference, but you should always expect the unexpected from nonlinearities. The upside is that the receiver design is still low cost.