11.pdf

Chapter 11

Mixers, Modulators

and Demodulators

At base, radio communication involves

translating information into radio form,

letting it travel for a time, and translating it

back again. Translating information into

radio form entails the process we call modu-

lation , and demodulation is its reverse. One

way or another, every transmitter used for

radio communication, from the simplest to

the most complex, includes a means of

modulation; one way or another, every re-

ceiver used for radio communication, from

the simplest to the most complex, includes

a means of demodulation.

Modulation involves varying one or

both of a radio signal’s basic characteris-

tics — amplitude and frequency (or phase)

— to convey information. A circuit, stage

or piece of hardware that modulates is

called a modulator .

Demodulation involves reconstructing

the transmitted information from the

changing characteristic(s) of a modulated

radio wave. A circuit, stage or piece of

hardware that demodulates is called a

demodulator .

Many radio transmitters, receivers and

transceivers also contain mixers — cir-

cuits, stages or pieces of hardware that

combine two or more signals to produce

additional signals at sums of and differ-

ences between the original frequencies.

Amateur Radio textbooks have tradition-

ally handled mixers separately from

modulators and demodulators, and modu-

lators separately from demodulators.

This chapter, by David Newkirk, ex-

W9VES, and Rick Karlquist, N6RK, exam-

ines mixers, modulators and demodulators

together because the job they do is essen-

tially the same. Modulators and demodu-

lators translate information into radio

form and back again; mixers translate one

frequency to others and back again. All of

these translation processes can be thought

of as forms of frequency translation or fre-

quency shifting — the function tradition-

ally ascribed to mixers. We’ll therefore

begin our investigation by examining what

a mixer is (and isn’t), and what a mixer

does.

to produce its output signal from instant to

instant ( Fig 11.2 ). Comparing the output

spectra of the combiner and mixer, we see

that the combiner’s output contains only

the frequencies of the two inputs, and

nothing else, while the mixer’s output

contains new frequencies. Because it com-

bines one energy with another, this pro-

cess is sometimes called heterodyning ,

from the Greek words for other and power .

Mixing as Multiplication

Since a mixer works by means of multi-

plication, a bit of math can show us how

they work. To begin with, we need to rep-

resent the two signals we’ll mix, A and B,

mathematically. Signal A’s instantaneous

a mplitude equals

THE MECHANISM OF MIXERS AND

MIXING

What is a Mixer?

Mixer is a traditional radio term for a

circuit that shifts one signal’s frequency up

or down by combining it with another sig-

nal. The word mixer is also used to refer to

a device used to blend multiple audio in-

puts together for recording, broadcast or

sound reinforcement. These two mixer

types differ in one very important way: A

radio mixer makes new frequencies out of

the frequencies put into it, and an audio

mixer does not.

a π (1)

in which A is peak amplitude, f is fre-

quency, and t is time. Likewise, B’s in-

s tantaneous amplitude equals

sin

b π (2)

Since our goal is to show that multiply-

ing two signals generates sum and differ-

ence frequencies, we can simplify these

signal definitions by assuming that the

peak amplitude of each is 1. The equation

f or Signal A then becomes

(

sin

Mixing Versus Adding

Radio mixers might be more accurately

called “frequency mixers” to distinguish

them from devices such as “microphone

mixers,” which are really just signal com-

biners , summers or adders . In their most

basic, ideal forms, both devices have two

inputs and one output. The combiner sim-

ply adds the instantaneous voltages of the

two signals together to produce the output

at each point in time ( Fig 11.1 ). The mixer,

on the other hand, multiplies the instanta-

neous voltages of the two signals together

)

π= (3)

a nd the equation for Signal B becomes

(

)

sin

)

π= (4)

Each of these equations represents a

sine wave and includes a subscript letter

to help us keep track of where the signals

go.

Merely combining Signal A and Signal

)

sin

Mixers, Modulators and Demodulators

11.1

Fig 11.1 — Adding or summing two sine waves of different frequencies (f1 and f2) combines their amplitudes without

affecting their frequencies. Viewed with an oscilloscope (a real-time graph of amplitude versus time), adding two signals

appears as a simple superimposition of one signal on the other. Viewed with a spectrum analyzer (a real-time graph of signal

amplitude versus frequency), adding two signals just sums their spectra. The signals merely coexist on a single cable or

wire. All frequencies that go into the adder come out of the adder, and no new signals are generated. Drawing B, a block

diagram of a summing circuit, emphasizes the stage’s mathematical operation rather than showing circuit components.

Drawing C shows a simple summing circuit, such as might be used to combine signals from two microphones. In audio work,

a circuit like this is often called a mixer — but it does not perform the same function as an RF mixer.

Fig 11.2 — Multiplying two sine waves of different frequencies produces a new output spectrum. Viewed with an oscilloscope,

the result of multiplying two signals is a composite wave that seems to have little in common with its components. A

spectrum-analyzer view of the same wave reveals why: The original signals disappear entirely and are replaced by two new

signals — at the sum and difference of the original signals’ frequencies. Drawing B diagrams a multiplier, known in radio

work as a mixer. The X emphasizes the stage’s mathematical operation. (The circled X is only one of several symbols you

may see used to represent mixers in block diagrams, as Fig 11.3 explains.) Drawing C shows a very simple multiplier circuit.

The diode, D, does the mixing. Because this circuit does other mathematical functions and adds them to the sum and

difference products, its output is more complex than f1 + f2 and f1 – f2, but these can be extracted from the output by

filtering.

11.2

Chapter 11

B by letting them travel on the same wire

develops nothing new:

() ()

in the many textbooks available on this fas-

cinating subject.) The difference between

the mixer we’ve been describing and any

mixer, modulator or demodulator that

you’ll ever use is that it’s ideal. We put in

two signals and got just two signals out.

Real mixers, modulators and demodula-

tors, on the other hand, also produce distor-

tion products that make their output spectra

“dirtier” or “less clean,” as well as putting

out some energy at input-signal frequen-

cies and their harmonics. Much of the art

and science of making good use of multi-

plication in mixing, modulation and de-

modulation goes into minimizing these

unwanted multiplication products (or their

effects) and making multipliers do their

frequency translations as efficiently as

possible.

( )

(

)

sin

(5)

As needlessly reflexive as equation 5

may seem, we include it to highlight the

fact that multiplying two signals is a quite

different story. From trigonometry, we

know that multiplying the sines of two

variables can be expanded according to the

relationship

[

(

)

(

)

]

sin

cos

−

cos

(6)

Conveniently, Signals A and B are both

sinusoidal, so we can use equation 6 to

determine what happens when we multi-

ply Signal A by Signal B. In our case, x =

f b t, so plugging them into

equation 6 gives us

f a t and y = 2

Putting Multiplication to Work

Piecing together a coherent picture of

how multiplication works in radio com-

munication isn’t made any easier by the

fact that traditional terms applied to a

given multiplication approach and its

products may vary with their application.

If, for instance, you’re familiar with stan-

dard textbook approaches to mixers,

modulators and demodulators, you may be

wondering why we didn’t begin by work-

ing out the math involved by examining

amplitude modulation , also known as AM .

“Why not tell them about the carrier and

how to get rid of it in a balanced modula-

tor ?” A transmitter enthusiast may ask

“Why didn’t you mention sidebands and

how we conserve spectrum space and

power by getting rid of one and putting all

of our power into the other?” A student of

radio receivers, on the other hand, expects

any discussion of the same underlying

multiplication issues to touch on the top-

ics of LO feedthrough , mixer balance

( single or double ?), image rejection and

so on.

You likely expect this book to spend

some time talking to you about these

things, so it will. But this radio-amateur-

oriented discussion of mixers, modulators

and demodulators will take a look at their

common underlying mechanism before

turning you loose on practical mixer,

modulator and demodulator circuits. Then

you’ll be able to tell the forest from the

trees. Fig 11.3 shows the block symbol for

a traditional mixer along with several IEC

symbols for other functions mixers may

perform.

It turns out that the mechanism underly-

ing multiplication, mixing, modulation

and demodulation is a pretty straightfor-

ward thing: Any circuit structure that

nonlinearly distorts ac waveforms acts as

a multiplier to some degree.

Fig 11.3 — We commonly symbolize

mixers with a circled X (A) out of

tradition, but other standards

sometimes prevail (B, C and D).

Although the converter/changer symbol

(D) can conceivably be used to indicate

frequency changing through mixing,

the three-terminal symbols are arguably

better for this job because they convey

the idea of two signal sources resulting

in a new frequency. (IEC stands for

International Electrotechnical

Commission.)

() ()

•

(

[

]

)

(

[

]

)

cos

−

cos

(7)

Now we see two momentous results: a

sine wave at the frequency difference be-

tween Signal A and Signal B 2π(f a – f b )t,

and a sine wave at the frequency sum of

Signal A and Signal B 2π(f a + f b )t. (The

products are cosine waves, but since

equivalent sine and cosine waves differ

only by a phase shift of 90°, both are called

sine waves by convention.)

This is the basic process by which we

translate information into radio form and

translate it back again. If we want to trans-

mit a 1-kHz audio tone by radio, we can

feed it into one of our mixer’s inputs and

feed an RF signal — say, 5995 kHz — into

the mixer’s other input. The result is two

radio signals: one at 5994 kHz (5995 – 1)

and another at 5996 kHz (5995 + 1). We

have achieved modulation.

Converting these two radio signals back

to audio is just as straightforward. All we

do is feed them into one input of another

mixer, and feed a 5995-kHz signal into the

mixer’s other input. Result: a 1-kHz tone.

We have achieved demodulation; we have

communicated by radio.

The key principle of a radio mixer is that

in mixing multiple signal voltages together,

it adds and subtracts their frequencies to

produce new frequencies. (In the field of

signal processing, this process, multiplica-

tion in the time domain, is recognized as

equivalent to the process of convolution in

the frequency domain. Those interested in

this alternative approach to describing the

generation of new frequencies through

mixing can find more information about it

Nonlinear Distortion?

The phrase nonlinear distortion sounds

redundant, but isn’t. Distortion, an exter-

nally imposed change in a waveform, can

be linear; that is, it can occur indepen-

dently of signal amplitude. Consider a

radio receiver front-end filter that passes

only signals between 6 and 8 MHz. It does

this by linearly distorting the single com-

plex waveform corresponding to the wide

RF spectrum present at the radio’s antenna

terminals, reducing the amplitudes of fre-

quency components below 6 MHz and

above 8 MHz relative to those between 6

and 8 MHz. (Considering multiple signals

on a wire as one complex waveform is just

as valid, and sometimes handier, than con-

sidering them as separate signals. In this

case, it’s a bit easier to think of distortion

as something that happens to a waveform

rather than something that happens to

separate signals relative to each other. It

would be just as valid — and certainly

more in keeping with the consensus view

— to say merely that the filter attenuates

signals at frequencies below 6 MHz and

above 8 MHz.) The filter’s output wave-

form certainly differs from its input wave-

form; the waveform has been distorted.

Mixers, Modulators and Demodulators

11.3

But because this distortion occurs inde-

pendently of signal level or polarity, the

distortion is linear. No new frequency

components are created; only the ampli-

tude relationships among the wave’s ex-

isting frequency components are altered.

This is amplitude or frequency distortion,

and all filters do it or they wouldn’t be

filters.

Phase or delay distortion , also linear,

causes a complex signal’s various compo-

nent frequencies to be delayed by differ-

ent amounts of time, depending on their

frequency but independently of their am-

plitude. No new frequency components

occur, and amplitude relationships among

existing frequency components are not

altered. Phase distortion occurs to some

degree in all real filters.

The waveform of a non-sinusoidal sig-

nal can be changed by passing it through a

circuit that has only linear distortion, but

only nonlinear distortion can change the

waveform of a simple sine wave. It can

also produce an output signal whose out-

put waveform changes as a function of the

input amplitude, something not possible

with linear distortion. Nonlinear circuits

often distort excessively with overly

strong signals, but the distortion can be a

complex function of the input level.

Nonlinear distortion may take the form

of harmonic distortion , in which integer

multiples of input frequencies occur, or

intermodulation distortion (IMD) , in

which different components multiply to

make new ones.

Any departure from absolute linearity

results in some form of nonlinear distor-

tion, and this distortion can work for us or

against us. Any so-called linear amplifier

distorts nonlinearly to some degree; any

device or circuit that distorts nonlinearly

can work as a mixer, modulator, demodu-

lator or frequency multiplier. An ampli-

fier optimized for linear operation will

nonetheless mix, but inefficiently; an

amplifier biased for nonlinear amplifica-

tion may be practically linear over a given

tiny portion of its input-signal range. The

trick is to use careful design and compo-

nent selection to maximize nonlinear dis-

tortion when we want it, and minimize it

when we don’t. Once we’ve decided to

maximize nonlinear distortion, the trick is

to minimize the distortion products we

don’t want, and maximize the products we

desire.

generating some unwanted IMD products

— spurious signals, or spurs — as they go.

Receivers are especially sensitive to un-

wanted mixer IMD because the signal-

level spread over which they must operate

without generating unwanted IMD is

often 90 dB or more, and includes infini-

tesimally weak signals in its span. In a re-

ceiver, IMD products so tiny that you’d

never notice them in a transmitted signal

can easily obliterate weak signals. This is

why receiver designers apply so much ef-

fort to achieving “high dynamic range.”

The degree to which a given mixer,

modulator or demodulator circuit pro-

duces unwanted IMD is often the reason

why we use it, or don’t use it, instead of

another circuit that does its wanted-IMD

job as well or even better.

Other Mixer Outputs

In addition to desired sum-and-differ-

ence products and unwanted IMD prod-

ucts, real mixers also put out some energy

at their input frequencies. Some mixer

implementations may suppress these

outputs — that is, reduce one or both of

their input signals by a factor of 100 to

1,000,000, or 20 to 60 dB. This is good

because it helps keep input signals at the

desired mixer-output sum or difference

frequency from showing up at the IF ter-

minal — an effect reflected in a receiver’s

IF rejection specification. Some mixer

types, especially those used in the

vacuum-tube era, suppress their input-sig-

nal outputs very little or not at all.

Input-signal suppression is part of an

overall picture called port-to-port isola-

tion . Mixer input and output connections

are traditionally called ports . By tradition,

the port to which we apply the shifting

signal is the local-oscillator (LO) port.

The convention for naming the other two

ports (one of which must be an output, and

the other of which must be an input) is

usually that the higher-frequency port is

called the RF (radio frequency) port and

the lower-frequency port is called the IF

(intermediate frequency) port. If a mixer’s

output frequency is lower than its input

frequency, then the RF port is an input and

the IF port is an output. If the output fre-

quency is higher than the input frequency,

the IF port may be the input and the RF

port may be the output. (We hedge with

may be because usage varies. When in

doubt, check a diagram carefully to deter-

mine which port is the “gozinta” and

which port is the “gozouta.”)

It’s generally a good idea to keep a

mixer’s input signals from appearing at its

output port because they represent energy

that we’d rather not pass on to subsequent

circuitry. It therefore follows that it’s usu-

Fig 11.4 — Feeding two signals into one

input of a mixer results in the same

output as if f 1 and f 2 are each first

mixed with f 3 in two separate mixers,

and the outputs of these mixers are

combined.

the same input by themselves or by each

other. (Multiplying a signal by itself —

squaring it — generates harmonic distor-

tion [specifically, second-harmonic distor-

tion] by adding the signal’s frequency to

itself per equation 7. Simultaneously squar-

ing two or more signals generates simulta-

neous harmonic and intermodulation

distortion, as we’ll see later when we ex-

plore how a diode demodulates AM.)

Consider what happens when a mixer

must handle signals at two different fre-

quencies (we’ll call them f 1 and f 2 ) ap-

plied to its first input, and a signal at a

third frequency (f 3 ) applied to its other

input. Ideally, a mixer multiplies f 1 by f 3

and f 2 by f 3 , but does not multiply f 1 and f 2

by each other . This produces output at the

sum and difference of f 1 and f 3 , and the

sum and difference of f 2 and f 3 , but not the

sum and difference of f 1 and f 2 . Fig 11.4

shows that feeding two signals into one

input of a mixer results in the same output

as if f 1 and f 2 are each first mixed with f 3

in two separate mixers, and the outputs of

these mixers are combined. This shows

that a mixer, even though constructed with

nonlinearly distorting components, actu-

ally behaves as a linear frequency shifter.

Traditionally, we refer to this process as

mixing and to its outputs as mixing prod-

ucts , but we may also call it frequency

conversion , referring to a device or circuit

that does it as a converter , and to its out-

puts as conversion products.

Real mixers, however, at best act only

as reasonably linear frequency shifters,

Keeping Unwanted Distortion Products

Down

Ideally, a mixer multiplies the signal at

one of its inputs by the signal at its other

input, but does not multiply a signal at the

same input by itself, or multiple signals at

11.4

Chapter 11

ally a good idea to keep a mixer’s LO-port

energy from appearing at its RF port, or its

RF-port energy from making it through to

the IF port. But there are some notable

exceptions.

rier, the upper sideband (USB) , and the

difference product, which comes out a fre-

quency lower than that of the carrier, the

lower sideband (LSB) . We have achieved

amplitude modulation.

Mixers and Amplitude Modulation

Now that we’ve just discussed what a

fine thing it is to have a mixer that doesn’t

let its input signals through to its output

port, we can explore a mixing approach that

outputs one of its input signals so strongly

that the fed-through signal’s amplitude at

least equals the combined amplitudes of the

system’s sum and difference products! This

system, amplitude modulation , is the old-

est means of translating information into

radio form and back again. It’s a frequency-

shifting system in which the original

unmodulated signal, traditionally called the

carrier , emerges from the mixer along with

the sum and difference products, tradition-

ally called sidebands .

We can easily make the carrier pop out

of our mixer along with the sidebands

merely by building enough dc level shift

into the information we want to mix so

that its waveform never goes negative.

Back at equations 1 and 2, we decided to

keep our mixer math relatively simple by

setting the peak voltage of our mixer’s

input signals directly equal to their sine

values. Each input signal’s peak voltage

therefore varies between +1 and –1, so all

we need to do to keep our modulating-

signal term (provided with a subscript m

to reflect its role as the modulating or in-

formation waveform) from going negative

is add 1 to it. Identifying the carrier term

with a subscript c, we can write

(

Why We Call It Amplitude Modulation

We call the modulation process de-

scribed in equation 8 amplitude modula-

tion because the complex waveform

consisting of the sum of the sidebands and

carrier varies with the information

signal’s magnitude (m). Concepts long

used to illustrate AM’s mechanism may

mislead us into thinking that the carrier

varies in strength with modulation, but

careful study of equation 9 shows that this

doesn’t happen. The carrier, sin 2πf c t,

goes into the modulator — we’re in the

modulation business now, so it’s fitting to

use the term modulator instead of mixer

— as a sinusoid with an unvarying maxi-

mum value of |1|. The modulator multi-

plies the carrier by the dc level (+1) that

we added to the information signal (m sin

2πf m t). Multiplying sin 2πf c t by 1 merely

returns sin 2

f c t. We have proven that the

carrier’s amplitude does not vary as a re-

sult of amplitude modulation. The carrier

is, however, used by many circuits as a

reference signal.

Overmodulation

Since the audio we are transmitting in

AM shows up entirely as energy in its side-

bands, it follows that the more energetic

we make the sidebands, the more informa-

tion energy will be available for an AM

receiver to “recover” when it demodulates

the signal. Even in an ideal modulator,

there’s a practical limit to how strong we

can make an AM signal’s sidebands rela-

tive to its carrier, however. Beyond that

limit, we severely distort the waveform we

want to translate into radio form.

We reach AM’s distortion-free modu-

lation limit when the sum of the sidebands

and carrier at the modulator output just

reaches zero at the modulating wave-

form’s most negative peak ( Fig 11.5 ). We

call this condition 100% modulation , and

it occurs when m equals 1. (We enumerate

modulation percentage in values from 0 to

100%. The lower the number, the less in-

formation energy in the sidebands. You

may also see modulation enumerated in

terms of a modulation factor from 0 to 1,

which directly equals m; a modulation

factor of 1 is the same as 100% modula-

tion.) Equation 9 shows that each side-

band’s voltage is half that of the carrier.

Power varies as the square of voltage, so

the power in each sideband of a 100%-

modulated signal is therefore ( 1 / 2 ) 2 times,

or 1 / 4 , that of the carrier. A transmitter

)

signal

sin

(8)

f m t) term

has company in the form of a coefficient,

m. This variable expresses the modulating

signal’s varying amplitude — variations

that ultimately result in amplitude modu-

lation. Expanding equation 8 according to

equation 6 gives us

Notice that the modulation (2

Fig 11.5 — Graphed in terms of

amplitude versus time (A), the envelope

of a properly modulated AM signal

exactly mirrors the shape of its

modulating waveform, which is a sine

wave in this example. This AM signal is

modulated as fully as it can be — 100%

— because its envelope just hits zero

on the modulating wave’s negative

peaks. Graphing the same AM signal in

terms of amplitude versus frequency

(B) reveals its three spectral

components: Carrier, upper sideband

and lower sideband. B shows

sidebands as single-frequency

components because the modulating

waveform is a sine wave. With a

complex modulating waveform, the

modulator’s sum and difference

products really do show up as bands

on either side of the carrier (C).

signal

sin

(

)

cos

−

(9)

(

) t

−

cos

The modulator’s output now includes

the carrier (sin 2

f c t) in addition to sum

and difference products that vary in

strength according to m. According to the

conventions of talking about modulation,

we call the sum product, which comes out

at a frequency higher than that of the car-

Mixers, Modulators and Demodulators

11.5

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