We will look at a mixing arrangement where the local oscillator runs at the frequency of the input signal yielding an IF of zero Hz.
The acronym "IF" as applied to superheterodyne receivers stands for "intermediate frequency" and so the above title is, to be a purist, an absurdity in referencing a "frequency frequency" but I've decided that I don't care. It's too easy to talk about an IF device or circuit or system or whatever so I will beg your indulgence in that for the time being.
The superheterodyne principle is that an input signal of whatever frequency is "mixed" with the frequency of a "local oscillator" to yield a new signal at an "intermediate frequency" which we call the IF. In a typical AM radio, the IF is 455 kHz while in a typical FM radio, the IF is 10.7 MHz. In both cases, the local oscillator runs at the frequency of the input signal but displaced by the IF. If you're listening to AM radio station WINS in New York City at 1010 kHz on your dial, the local oscillator will be running at 1465 kHz.
Here, however, we will look at a mixing arrangement where the local oscillator runs at the frequency of the input signal yielding an IF of zero Hz, hence the above title.
The following diagram is a zero frequency IF stage for use in superheterodyne reception where input #1 is passed along to the output so long as its frequency wm = 2 x pi x fm is close enough to a second local oscillator frequency wc = 2 x pi x fc. How close the input frequency must be is set by a pair of lowpass filters. The lower the cutoff frequency of the two lowpass filters, the narrower the selectivity becomes.
Figure 1 Zero frequency IF and equations, 1st examination
The ideal multipliers are our mixers. Their operation as shown above is based on the following trig identity: cos a x cos b = ½ x (cos (a+b) + cos (a-b))
However, the algebra can be looked at in an alternative fashion. We can let "a" represent "wc" and "b" represent "wm", or we can go the other way around. It doesn't matter.
Figure 2 Zero frequency IF and equations, 2nd examination
In this case, we use the following equivalent trig identity: cos b x cos a = ½(cos (b+a) + cos (b-a))
This is a difference which makes no difference. The end results of Figure 1 and Figure 2 are the same. A zero frequency IF simulation in SPICE looks like this:
Figure 3 Zero frequency IF simulation with approximately 5 kHz lowpass filters
... or go with a narrower bandpass, like this:
Figure 4 Zero frequency IF simulation with approximately 500 Hz lowpass filters
The overall bandpass is twice the lowpass filter cutoff frequency. Using a local oscillator frequency of 1 MHz and for the single RC filters shown, the calculated bandpasses are as follows:
Figure 5 Zero frequency IF bandpasses
Note the ten to one ratio of bandwidths versus the ten to one ratio of lowpass filter cutoff frequencies which are 1/(2 pi x 3160 x 0.01E-6) = 5037 Hz ...and... 1/(2 pi x 3160 x 0.1E-6) = 503.7 Hz.
This technique was employed in Building Block 38 (BB38) of the US Navy CVA VAST test system. BB38 was called the low frequency wave analyzer. The zero frequency IF wasn't originated there however. The method had been copied from the now discontinued Hewlett-Packard HP3590A analyzer.
There was however, just one "gotcha." The quartet of mixers could be disrupted by DC offsets. To avoid that problem. the lowpass filters were DC blocked with extremely low frequency highpass corners. As a result, the IF bandpasses as seen above had an infinitely deep but very narrow notch right at the center frequency.
John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).