## RE: [softrock40] Rant on image rejection.

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• I am .... Im not the dullest knife in the drawer ... it takes me longer than most to cut through but I can. hi:hi... Mike KC7NOA To: softrock40@yahoogroups.com
Message 1 of 21 , Dec 22, 2011
I am ....
Im not the dullest knife in the drawer ... it takes me longer than most to cut through but I can. hi:hi...

Mike KC7NOA

To: softrock40@yahoogroups.com
From: MertNellis@...
Date: Thu, 22 Dec 2011 16:21:44 -0600
Subject: Re: [softrock40] Rant on image rejection.

Hi Niels,
Thanks for the good effort on explaining some of the signal processing in the softrock system.  I will copy your Rant and study the equations more closely.  I have been through some of it before and am trying to get my mind firmly around the concepts.  You will help.  I have not seen other comments on your Rant and suspect that most are not interested in that much much math.
Thank you and 73  Mert  W0UFO
----- Original Message -----
From: Niels
Sent: Tuesday, December 20, 2011 6:17 PM
Subject: [softrock40] Rant on image rejection.

Hi all,

I wrote this in response to the image rejection posts that frequently
appear on this list.

== Introduction ==

Almost all the articles in ham radio literature explain image rejection
and other "I/Q" signal processing by treating I and Q signals
separately. I feel that part of the picture is missing when it is
explained that way, as mathematically they are conceptually closely
linked together. In fact, they constitute a single complex-valued, i.e.
two-dimensional, signal.

Unfortunately, many hams are not familiar with complex-valued numbers
and/or signal processing. As it takes effort to learn something new,
especially when it concerns mathematics, many leave the subject alone.
This is made worse by the term "complex" numbers. That is why I prefer
to use the term complex-valued numbers.

== Complex-valued numbers ==

I am convinced that learning about complex numbers and some of the
mathematics involved is very useful -- if not almost essential -- when
trying to fully understand the workings of quadrature down-converter
front-ends, such as the Softrock series. In fact, using complex-valued
mathematics, the Softrock radio architecture can be described by one
simple equation:

y(t) = x(t)*exp(-j*2*pi*f*t)

,where 'x(t)' is the antenna RF signal, 'f' is the oscillator frequency
(in Hz), 't' is time (in seconds), 'y(t)' is a single signal
representing 'I' & 'Q' and 'j' has a special meaning in complex-valued
mathematics. That's it! no need to remember trigonometric identities,
and -- if you're familiar with complex-valued mathematics -- completely
obvious that this is a down-converter and that it has infinite image
rejection.

All quadrature down-converters attempt to implement the above equation.
The amount of image rejection achieved by a real-world down-converter
depends on how accurately it approaches that equation.

A second advantage of knowing how to do complex-valued mathematics is
that there is no need to remember many of the trigonometric identities;
one can derive them in a few simple steps.

== Double-sided spectrum ==

Consider the spectrum of the antenna signal; Normally, the frequency
axis starts at DC, i.e. 0 Hz, and extends into infinity. In effect, the
frequency axis has a "start" but no ending; this is a problem when
shifting the spectrum! One could say that a down-converter shifts the
spectrum "to the left", i.e. towards DC. But .. where does the original
spectrum around DC go? To avoid this dilemma, I prefer to consider the
double-sided spectrum.

The double-sided spectrum starts at a frequency of -infinity, has 0 Hz
in the middle, and extends to +infinity. As there is no "start" or "end"
to the spectrum, one can shift it to the left (down-conversion) or right
(up-conversion) without "losing" parts of it. I find that a more
comfortable point-of-view when dealing with up/down-conversion than
fiddling with a single-sided spectrum.

== Complex-valued operations and the double-sided spectrum ==

Without proof, here are some of the most relevant mathematical rules:

1) multiply by exp(-j*2*pi*f*t)  =>  shift the entire double-sided
spectrum by 'f' Hz to the left.
2) multiply by exp(j*2*pi*f*t)  =>  shift the entire double-sided
spectrum by 'f' Hz to the right.
3) add two signals x1(t) + x2(t)  =>  the spectra of x1 and x2 simply add.
4) the double-sided spectrum of a, so called, real signal (not
complex-valued), is symmetrical in amplitude around 0 Hz.
5) the double-sided spectrum of a, so called, real signal (not
complex-valued), is anti-symmetrical in phase around 0 Hz.
6) when a double-sided spectrum is not symmetrical in amplitude around 0
Hz, it cannot be represented by a real signal.
7) if a complex-valued signal is made non-complex, i.e. real, the
negative part responses will appear in the positive half and vice-versa.

Note: all of the above rules can be proven by invoking the Fourier
transform.

I hope I haven't scared you off! Here are some of the practical

== Practical implications ==

The RF antenna signal is a real signal -- it cannot be complex-valued as
they only exist in pairs, i.e. two signals. As such, its double-sided
spectrum is symmetrical (rule 4). This basically means that if there is
a signal at 10 MHz, the same signal will exist at -10 MHz.

The purpose of the Softrock is to shift down the spectrum of interest to
around DC so that the PC soundcard can digitize it for further
processing. Lets assume we want to receive the 10 MHz. The Softrock must
shift the spectrum by an amount that will put the 10 MHz signal within
the limited audio bandwidth of the soundcard. Therefore, we set the
Softrock LO frequency to f=9.99 MHz, for example.

The Softrock implements a multiplication by exp(-j*2*pi*f*t), which
shifts the entire double-sided spectrum to the left (rule 1). When we
shift the double-sided spectrum to the left (down-conversion) it will no
longer be symmetrical, as the -10 MHz signal will appear at -19.99 MHz
and the 10 MHz signal will appear at 0.01 MHz! This situation CANNOT be
represented by a real signal -- it MUST be complex-valued (rule 6).
Hence, we need a pair of voltage signals "I" and "Q".

When a Softrock isn't wired correctly, either "I" or "Q" will have zero
volts and image rejection is lost. The Softrock no longer performs the
multiplication with the complex-valued exponential exp(-j*2*pi*f*t),
which has only a response at -f Hz, given a double-sided spectrum. In
the case that "Q" is zero, an additional term exp(j*2*pi*f*t) will be
introduced (rule 7):

y(t) = x(t)*[exp(-j*2*pi*f*t) + exp(j*2*pi*f*t)]

,which makes the Softrock both an up-converter and a down-converter
(rules 1 and 2), in terms of the double-sided spectrum. In such a case,
a 10 MHz RF signal (appearing at -10 MHz and 10 MHz in the double-sided
antenna signal spectrum) will be converted to:

-10.0 - 9.99 = -19.99 (down-conversion negative part)
-10.0 + 9.99 = -0.01 (up-conversion negative part)
10.0 - 9.99 = 0.01 (down-conversion positive part)
10.0 + 9.99 = 19.99 (up-conversion positive part)

Note that everything is symmetrical around DC; it is a real signal.
Doing the same for a 9.98 MHz signal:

-9.98 - 9.99 =  -19.97
-9.98 + 9.99 = 0.01
9.98 - 9.99 = -0.01
9.98 + 9.99 = 19.97

Note that there is no image rejection; 9.98 and 10 MHz both end up at
+/- 0.01 MHz.

== /rant ==

The above has barely touched on the subject of complex-valued math and
its uses in radio applications. I hope you've found this useful, if not,
sorry for the bandwidth.

73,
Niels PA1DSP.

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