+1
With the most practical option of AC coupling the IF port (or high-pass
filtering in software), you'd get a double-hump response to a single
tone. With great effort you might get one of the humps suppressed by
60dB, or greatly narrow the notch around DC, and that might make it a
useful instrument, but it will always look a bit shabby to someone used
to a commercial spectrum analyser. Still, it would be a project worth
building.
If trying to DC-couple, then there are a lot of other challenges such as
harmonics of the input signal pulling the LO (if it is a PLL), LO signal
getting into the input port via the mixer, then reflecting off your DUT,
giving you a VSWR-dependent DC signal that is indistinguishable from an
input signal, etc. They are mostly solvable in principle with brute
force heroic implementation that costs as much as a superhet. One quite
nasty problem is 1/f noise in the baseband amplifiers - that is hard to
fix or prevent.
Another problem with direct conversion is the response to harmonics of
the LO. If you want low noise, stable gain with time and temperature,
and linearity with variations in the input amplitude (to avoid IMD
between the incoming tones) then you want to drive the LO port of the
mixer hard, with a square wave. It will naturally respond to the wanted
signal (let's call it f_displayed) but it will also respond to
3*f_displated, 5*f_displayed etc. so, for a single tone input, you will
see the wanted tone and all of its subharmonics with diminishing
amplitude at the lower subharmonics. Perhaps you could to a first order
subtract them out in software, (e.g. if you have a tone at 15MHz,
measure it at 15MHz LO, then subtract a third of that out when your
sweep is at 5MHz, and subtract out a fifth of the 15MHz measurement when
your LO is at 3MHz.) This won't work all that well, so perhaps the best
approach is to use it as a narrow-band instrument (covering less than an
octave at a time) and put a band-pass filter ahead of it.
I did once consider that for a direct-conversion radio covering say
0-100MHz, you might be able to generate a hard-switched LO that is a
1-bit sigma-delta approximation to a pure sine-wave with practically no
content at the odd harmonics of the wanted LO frequency, (but lots of
noisy stuff above maybe 500MHz). E.g. get a FPGA with a serdes and make
it generate an 8Gbps bitstream that, if low-pass filtered, would give a
pure sine wave at the wanted LO frequency e.g. 1MHz. Instead of low-pass
filtering the bitstream, instead pipe it straight from the fpga serdes
into the LO port of a mixer like the AD8343 or LT5560. Then the mixer
should respond to the wanted 1MHz input signal but not to 3MHz, 5MHz,
7MHz, etc. input signals. There would be a response to all of the
pseudo-noise hash that the bitstream contains above 500MHz, so you would
need a low-pass filter on the RF input, which should be feasible.
Also, consider the option of buying a $10 DVB-T USB dongle and using
that instead. There is existing software for it (although I have not
used it). In this case the dynamic range is probably quite limited due
to the 8-bit converters, but the price is right.
Chris