The Digital Down Converter (DDC) is a basic block of FPGA-implemented SDR radios. Per Wikipedia:

Below is an illustration of a DDC stage (A digital down-converter (DDC) converts a digitized, band limited signal to a lower frequency signal at a lower sampling rate in order to simplify the subsequent radio stages.

**don't worry about the math**-- it will be explained later in this post):

(Click on image to enlarge)

Note that,

**, the equations assume a simple cosine as the input signal.**

*for demonstration purposes***This post will look at the math involved in the Mixer**. I will leave the low-pass filter and sample-rate conversion for later posts.

First, some basics...

__Basic Down-Conversion Mixing:__Historically, down-conversion has been done in the analog domain with a "real" oscillator signal (e.g. cos(ω

_{o}t)). Let's first look at this type of basic mixer before tackling mixing in the Digital (and Complex) domain.

For demonstration purposes, I'll simplify the receive-signal to be a just simple cosine signal at a single frequency: cos(ω

_{i}t).

This mixer's math is straightforward. Let the output of the mixer be y(t).

y(t) = cos(ω

_{i}t)*cos(ω_{o}t)
We can use Trigonometric identities to express this equation in terms of sums and differences of frequencies:

y(t) = (cos((ω

_{i}−ω_{o})t))/2 + (cos((ω_{i}+ω_{o})t))/2**to base-band: both the desired signal**

*directly**and its image*will appear in the base-band spectrum (which, for the typical shortwave receiver, is the audio spectrum we want to hear).

This type of conversion is known as "Direct Conversion." For an example, see Rick Campbell's R1 receiver.

Visually, here's how the frequencies shift, shown as a

*single-sided*spectrum (which one would see if one did a "Trigonometric" Fourier Transform on the RF input. Refer to equation 27 in this paper by Robert Marcus for more detail)::

Note the overlap of the two signals down at base-band...our "desired" signal would be interfered with by any signal at its "image" on the other-side of the oscillator's frequency!

And if we were to consider the Mixer's input to be a

*band*of frequencies:

You can see the spectrum fold-back with the result being the desired signal and its undesired image occupying the same base-band frequencies. A single-sided spectrum graph has no negative frequencies, and any "calculated" negative frequencies become positive frequencies via the Trig identity: cos(−ω

_{i}t) = cos(ω

_{i}t).

We can get around this image problem by working in the domain of Complex Numbers (numbers with a

*real*and an

*imaginary*component). Specifically, we will use exponentials with complex arguments.

__Complex Exponentials -- A Quick Review of Some Important Identities:__Euler's Formula:

e

^{jθ}= cos(θ) + jsin(θ)Similarly,

e

^{−jθ}= cos(θ) - jsin(θ)
cos(θ) = (e

^{jθ}+ e^{−jθ})/2
sin(θ) = j(e

^{−jθ}− e^{jθ})/2
Let's consider a real signal at a single frequency f

Using the identities above this signal can be expressed in exponential form:

Below is a representation, in the complex frequency-domain, of this signal :

_{o}(Hz), and let's express this signal simply as a cosine: cos(2πf_{o}t).Using the identities above this signal can be expressed in exponential form:

cos(2πf

_{o}t) = (e^{j2πfot}+ e^{−j2πfot})/2
And, for completeness, here are a two more illustrations:

First, to complement the cosine representation above, here is a complex frequency-domain representation of a sine wave signal of the same frequency:

Second, we can use both of these illustrations of the signals in the complex frequency-domain to generate a visual representation of Euler's Formula, showing how the imaginary sine terms are rotated (when multiplied by j) to the real axis and then added to the real cosine terms:

Note that multiplying sin(2πf

First, to complement the cosine representation above, here is a complex frequency-domain representation of a sine wave signal of the same frequency:

Second, we can use both of these illustrations of the signals in the complex frequency-domain to generate a visual representation of Euler's Formula, showing how the imaginary sine terms are rotated (when multiplied by j) to the real axis and then added to the real cosine terms:

(Click on image to enlarge)

Note that multiplying sin(2πf

_{o}t) by j rotates the sine's complex frequency-domain imaginary components "clockwise" by 90 degrees to the real plane. If we were to instead multiply sin(2πf_{o}t) by -j, the rotation would be 90 degrees counter-clockwise, with the results again being on the real plane, but pointing in the opposite direction, and our final result (after adding the real components together) would represent Euler's Formula for a negative frequency, -f_{o}, instead of f_{o}.

__Frequency Conversion using Complex Numbers in Exponential Form:__Let's use the above equations to shift signal frequencies.

First, for ease of demonstration, let's define an

**input RF signal,**x(t),

**to be a simple cosine at a single frequency, f**

_{i }Hz. I.e.:

x(t) = cos(2πf

_{i}t)To minimize my typing, I will express this signal's frequency as its radian equivalent, where ω

_{i}= 2πf

_{i}.

x(t) = cos(ω

_{i}t)Using the identities above we can express cos(ω

_{i}t) as the sum of two complex exponential numbers:

cos(ω

_{i}t) = (e^{jωit}+ e^{−jωit})/2Now let's now define our oscillator's signal as a complex exponential:

e

^{jωot}(Note that from Euler's Formula e

^{jωot }= cos(ω

_{o}t) + jsin(ω

_{o}t)).

Our next step is to multiply together the input signal and the oscillator signal. We will call the output y(t).

y(t) = cos(ω

_{i}t) * e^{jωot}Replacing the input signal with its equivalent complex exponential form, the output equation becomes:

y(t) = [(e

^{jωit}+ e^{−jωit})/2]*e^{jωot}Finally, let's finish the multiplying. The result is:

y(t) = (e

^{j(ωi+ωo)t}+ e^{j(−ωi+ωo)t})/2Note that the frequency components in the arguments of both exponentials have been shift

*up*by ω

_{o.}

By the way, we can use Euler's Formula to express this result in terms of sines and cosines:

y(t) = [cos((ω

_{i}+ω_{o})t) + jsin((ω_{i}+ω_{o})t)]/2 + [cos((-ω_{i}+ω_{o})t) + jsin((-ω_{i}+ω_{o})t)]/2You can also derive the same result using only sines, cosines, and trigonometric identities and not complex exponentials, but the math is more cumbersome.

OK, so an NCO's output that is a complex-exponential with an argument whose frequency is +ω

_{o }will shift the incoming signal's frequency

__up__by ω

_{o}. What happens if the NCO's frequency is

*negative*, i.e. -ω

_{o}?

Using the same approach as above, the DDC's output can be shown to be:

y(t) = (e

In other words, both of the input signal's complex exponentials are shifted ^{j(ωi−ωo)t}+ e^{j(−ωi−ωo)t})/2*down*in frequency by ω

_{o}.

*Voila*, a Digital Mixer!

Again, note that y(t) can be expressed as the sum of sines and cosines. Using the identities at the start of this post:

y(t) = [cos((ω

_{i}-ω_{o})t) + jsin((ω_{i}-ω_{o})t)]/2 + [cos((ω_{i}+ω_{o})t) - jsin((ω_{i}+ω_{o})t)]/2^{j(ωi−ωo)t}+ e

^{j(−ωi−ωo)t})/2.

__Mixing with Complex Exponentials:__With a view towards understanding SDR building blocks, let's look at a Digital Mixer using the down-conversion math I've just discussed, and which shifts signal frequencies in the

*digital*(rather than analog) realm.

_{o}t):

Note that the oscillator's output is a

*complex exponential*with a

**negative**frequency in its argument. This negative frequency, as I've discussed earlier in this post, will

**down-convert**incoming RF.

The input signals' spectrum (shown below as

*two-sided,*with negative frequencies, per the frequencies in the complex arguments in the exponential form of the input signal) will shift down:

(Click on image to enlarge)

The illustration below shows a more general case for a band of frequencies at the DDC's input:

(Click on image to enlarge)

Note that, unlike the analog mixer described at the start of this post, this spectrum has both positive

*and*negative frequencies. In other words, we do not (yet) have an issue with "image wrapping" while we remain in the complex domain.

Another take on the same topic, from "Complex Signal Processing is Not -- Complex," by Ken Martin, showing the same:

(Source: "Complex Signal Processing is Not -- Complex,",Ken Martin)

Note (again) that the input consists of only real, not complex, numbers, but the output Y(t) will be in complex form, with both real and imaginary components.

Regarding the

**mixer (c)**in the image above, we can express its two outputs (

_{r}(t) and y

_{q}(t)) in terms of sines and cosines.

First, for simplicity of analysis let's define the input x

_{r}(t) to be a simple cosine at frequency ω

_{r}:

x(t) = cos(ω

_{r}t)

Noting that the oscillator frequency is ω

_{i}in this example, the output y

_{r}(t) is therefore:

_{r}(t) = cos(ω

_{r}t)*cos(ω

_{i}t)

which, using trig identities, expands to:

y

_{r}(t) = cos((ω_{r}−ω_{i})t)/2 + cos((ω_{r}+ω_{i})t)/2Now let's do the same to derive y

_{q}(t), the imaginary term...noting that −sin(ω

_{i}t) = sin(−ω

_{i}t), y

_{q}(t) becomes:

y

_{q}(t) = sin((ω_{r}−ω_{i})t)/2 − sin((ω_{r}+ω_{i})t)/2Is this result equivalent to the result derived (above) using complex-exponential equations?

Essentially, yes it is!

Let's multiply the y

_{q}(t) channel by j (i.e. add a phase shift of 90 degrees with, say, a Hilbert Transform) and call this output y

_{q}(t)' (note the 'prime'). Its equation is:

y

_{q}(t)' = jsin((ω_{r}−ω_{i})t)/2 − jsin((ω_{r}+ω_{i})t)/2_{r}(t) and y

_{q}(t)', the result is:

_{r}(t) + y

_{q}(t)' = [cos((ω

_{r}−ω

_{i})t) + jsin((ω

_{r}−ω

_{i})t)]/2 + [cos((ω

_{r}+ω

_{i})t) − jsin((ω

_{r}+ω

_{i})t)]/2

Which, from Euler's Formula, we know is exactly equivalent to:

_{r}(t) + y

_{q}(t)' = (e

^{j(ωr−ωi)t}+ e

^{j(−ωr−ωi)t})/2.

and which is exactly equivalent in form to the equation we derived earlier:

y(t) = (e

^{j(ωi−ωo)t}+ e^{j(−ωi−ωo)t})/2_{r}(t) and y

_{q}(t) are the fundamental mathematical elements (when the sine terms are multiplied by j) that constitute the Digital Mixer's operation when expressed in complex-exponential form.

__Links and Other Resources:__
An Excellent Tutorial on Complex Numbers: Quadrature Signals: Complex, but not Complicated, Richard Lyons

An interesting site to surf:

https://www.dsprelated.com/freebooks/mdft/Positive_Negative_Frequencies.html

An interesting site to surf:

https://www.dsprelated.com/freebooks/mdft/Positive_Negative_Frequencies.html

Representing a signal as the sum of two exponentials: http://ece.mst.edu/media/academic/ece/documents/classexp/ee216labs/EE216_Lab4.pdf

__The Significance of Negative Frequencies in Spectrum Analysis__, Robert Marcus, IEEE Transactions on Electromagnetic Compatibility, Dec. 1967. (A copy can be found here).

Good tutorial on Complex Number in Radio applications: Complex Signal Processing is Not -- Complex, Ken Martin

__Standard Caveat:__
I could have easily made a mistake in any of the above text, equations, or drawings. If something looks wrong or is confusing, please feel free to contact me. Thanks!

## No comments:

Post a Comment