Adjacent-Channel Interference for the AM Radio Design

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, A_icos(2πf_it). The received signal is now of the form

with f_i assumed to lie just outside the +/– 5-kHz channel bandwidth centered on f_c.

Using a generalized version of the phasor addition formula, you can show that the received envelope with single-tone interference is as follows:

Note that

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

where

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 f_c in the interference term:

In the formula A₁(t) = A_c[1 + am(t)],

Now, calculate

Add these complex numbers in rectangular form and then find the magnitude:

The last line follows from

Because

you can combine the terms and drop the absolute value. As a check, if A_i = 0, that is, no interference, the result for R(t) reduces to

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 A_i = 0, R(t) contains an absolute value. At this point, assume m(t) = cos(2πf_mt) 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 P_R(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

Also set A_m = 0.5, which is equivalent to setting a = 0.5 (50 percent modulation depth). The value of A_i steps over 0, 0.1, and 1.0. The 2-kHz message is within the 5-kHz message bandwidth requirement and

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

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.