Bandpass Lowpass Equivalent

7/29/2019 Bandpass Lowpass Equivalent

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1/26

II-1: Bandpass SignalsEquivalent Lowpass Signals

I&Q Signals

(PM-6.4.3)

Also called:

Equiv. Baseband Signals



2/26

Definition: Bandpass SignalA Bandpass Signal is a signal x(t ) whose Fourier transform X ( f )

is nonzero only in some small band around some “central”frequency f o.

The bandwidth B of the bandpass signal = the width of the positive-frequency

interval on which the signal is nonzero.

(Note: this is consistent with the bandwidth definition for lowpass signals).

f o− W − f o+W − f o− W f

| X ( f )|

f o f o+W − f o B

For example:

X ( f ) = 0 for | f - f o| > W where W < f o.



3/26

| X ( f )|

| X ( f )| f

f

f o

f o

− f o

− f o

Definition: Bandpass Signal (cont.)

Note that the choice of f o is arbitrary:

Bandpass signals are encountered when receiving radio

frequency (RF) signals such as communication and radar signals.In the analysis and actual processing of BP signals it is

convenient to work with a related, equivalent signal called the

Equivalent Lowpass Signal. This is a natural generalization of the idea of phasor used in sophomore-level circuits.



4/26

Recall: Phasor Idea Used in CircuitsIdea: Replace Acos(2π f o t +θ) by complex DC value Ae jθ

f o f

X a( f )

gets represented by a complex-valued signal

called the analytic signal:

xa(t ) = A exp{ j(2π f o t +θ)}

= A cos(2π f o t +θ) + j A sin(2π f o t +θ)

f

X l( f )

Then to get the phasor, we frequency-shift the

analytic signal down by f o to get:

xl = exp{– j2π f ot } xa(t )= Ae jθ

First, the sinusoid

f o− f o f

X ( f )

)2()2(

22

)2cos()(

θ π θ π

θ π

+−+ +=

+=

t f jt f j

o

oo e A

e A

t f At x



5/26

Recall: Phasor Idea Used in Circuits (cont.) phasor = equivalent lowpass signal representing the sinusoid

(that’s why we used the subscript l – for lowpass).

Note that this equivalent lowpass signal is complex valued,

whereas the bandpass signal (the sinusoid) it represents is

real valued.

f o f

X a( f )

f

X l( f )

Alternate View – Frequency Domain:

1. Suppress the negative frequency part of the sinusoid:

2. Frequency-shift the positive frequency part down to DC:



6/26

Frequency-Domain View of Equiv. LP Signal Now… use this FD view to do the same thing for a general bandpass

signal that consists of more than one frequency.….Then after that we interpret the results in the time domain.

Now to get the FT of the so-called Analytic Signal we

suppress the negative frequencies:

f o f

X a( f ) Note: Since | X a( f )| is

NOT even-symmetric,

the TD signal xa(t ) is

complex-valued.(see Porat p. 12, #9)

f o− f o f

X ( f )Bandpass Signal’s Fourier Transform:



7/26

F-D View of Equiv. LP Signal (cont.)

System View of Generating Analytic Signal: define a system

frequency response H ( f ) such that

⎪⎩

⎪

⎨

⎧

<+=

>−

=000

0

)( f j f

f j

f H then )(ˆ)(

)()()()(

f X j f X

f X f jH f X f X a

+=

+=

)()()(ˆ f X f H f X =where

H ( f )Σ

j

X ( f ) X a( f )

Called Hilbert Transformer



8/26

F-D View of Equiv. LP Signal (cont.)

Then to get the FT of the Equivalent Lowpass Signal, frequency-

shift the analytic signal down by f o to get:

X l( f )

f

Note that because | X l( f )| does not necessarily have even symmetry,

the equivalent lowpass signal is complex valued,

….. whereas the bandpass signal it represents is real valued.

Now… how do we describe the ELP signal in the Time-Domain?



9/26

T-D View of ELP Signal

Consider the IFT of H ( f ) X ( f ): )}()({)}(ˆ{)(ˆ 11 f X f H f X t x

−− == F F

)(ˆ)()( t x jt xt xa += (S)

Let xl(t ) be the time-domain signal that corresponds to X l( f ).Because it is the frequency-shifted version of xa(t ) …. using the

frequency-shift property of FT gives:

)()(2

t xet x at f j

loπ −= (º)

(Note: this is the same as an equation above for the phasor case!)

This, can then be written as:

[ ])(ˆ)()(2

t x jt xet xt f j

lo

+=

− π



10/26

I&Q Form of ELP Signal

An extremely useful viewpoint for the ELP signal is the I&Q form:

… since xl(t ) is complex-valued (see comment above in frequency-

domain discussion), we can write its real and imaginary parts, which

we will denote as )()()( t jxt xt x qil += (h)

where subscripts i and q are for In-phase (I) and Quadrature (Q).

We’d now like to find relationships between the bandpass signal x(t )

and the I-Q components of the lowpass equivalent signal.



11/26

Relationship: I&Q Parts and BP Signal

Solving (º) for the analytic signal gives

)()(2

t xet x lt f j

aoπ = ()

(Makes sense… xa(t ) is xl(t ) shifted up.)

[ ][ ][ ][ ]

[ ])(ˆ)(

)2cos()()2sin()(

)2sin()()2cos()(

)()()2sin()2cos(

)()()(2

t x jt x

t f t xt f t x j

t f t xt f t x

t jxt xt f jt f

t jxt xet x

oqoi

oqoi

qioo

qit f j

ao

+=

++

−=

++=

+=

π π

π π

π π

π

By (S)

Using the I-Q form given in (h) gives:



12/26

This shows how the I&Q components are related to the BP signal:

)2sin()()2cos()()( t f t xt f t xt x oqoi π π −=

Relationship: I&Q Parts and BP Signal (cont.)

(U)

Similarly – but less important – we have:

)2cos()()2sin()()(ˆ t f t xt f t xt x oqoi π π +=



13/26

Envelope/Phase Form of ELP Signal

0)()()( 22 ≥+= t xt xt A qi

where…

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=

)(

)(arctan)(θ

t x

t xt

i

q

This is an alternate form (but equally important to IQ form) of the

ELP signal. Note in (h) that the I&Q form is a “rectangular form”

for the complex ELP signal.

)()()( t jl et At x

θ = (²)

Note Similarityto Phasor!!

But…

Time-varying

Envelope &

Magnitude

So… converting to a “polar form” gives:



14/26

Often we need to convert between the two forms (rect & polar).

If in (²) we expand the complex exponential:

Relationship: Env/Phase and I&Q

)()(

)(

)](sin[)()](cos[)(

)()(

t xt x

t jl

qi

t t A jt t A

et At x

θ θ

θ

+=

=

By (h)

)](sin[)()(

)](cos[)()(

t t At x

t t At x

q

i

θ

θ

=

=

Thus….



15/26

We already saw Env/Phase form for the ELP signal…

Do we get something similar for the original BP signal ??

Envelope/Phase Form of BP Signal

)(ˆ)(

)](2sin[)()](2cos[)(

)(

])([)(

)](2[

)(2

t x jt x

t t f t jAt t f t A

et A

et Aet x

oo

t t f j

t jt f ja

o

o

+=

+++=

=

=

+

θ π θ π

θ π

θ π

By (S)

)](2cos[)()( t t f t At xo

θ π += (♣)

Using () and (²) we can write



16/26

So what we have just shown is:

Any BP signal can be expressed as:

x( t) = A( t) cos[2 f o t + ( t)]

Envelope/Phase Form of BP Signal (cont.)

)()()( t jl et At x

θ =

The LPE signal has the same envelope and phase as the BP signal

… compare (♣) and (²).

where A( t) 0.

Note: A( t) and ( t) vary slowly compared to cos(2 f o t).

A l G i f I&Q C



17/26

Analog Generation of I&Q Components

So… given the BP signal

)](2cos[)()( t t f t At x o θ π +=

we need to be able to extract through processing the I&Q signals:

)](sin[)()(

)](cos[)()(

t t At x

t t At x

q

i

θ

θ

=

=

As stated earlier… processing for radar & communication is actuallyimplemented using the ELP signal.

• Thus we need some way to get the ELP signal from a received

BP signal…

• The I&Q form is the most commonly used

A l G ti f I&Q C t ( t )



18/26

Analog Generation of I&Q Components (cont.)

These give the clue as to how to extract the I-Q signals by

using analog techniques. Using trigonometric identities:

[ ]

oi f t x

o

t x

ooo

t t f t At t A

t f t t f t At f t x

2atcentered but....)()(

)]()2(2cos[)()](cos[)(

)2cos())(2cos()(2)2cos()(2

θ π θ

π θ π π

++=

+=

f o f 2 f o0-f o-2 f o

A l G ti f I&Q C t ( t )



19/26

Analog Generation of I&Q Components (cont.)Similarly, we also get….

[ ]

oq f t x

o

t x

ooo

t t f t At t A

t f t t f t At f t x

2atcentered but....)(ˆ)(

)]()2(2sin[)()](sin[)(

)2sin())(2cos()(2)2sin()(2

θ π θ

π θ π π

++=

+=

Osc.90º

LPF

LPF

f o

xi(t )

x(t )

xq(t )

2cos(2π f ot )

-2sin(2π f ot )

Analog Circuitry to Generate I&Q

Di it l G ti f I&Q C t



20/26

Digital Generation of I&Q Components

Analog Devices

One Technology Way

PO Box 9106 Norwood, MA 02062-9106

www.analog.com

U f Th Id



21/26

Uses of These Ideas

• Bandpass Signal Model

– usually used to model RF signals in radar and communications

– also often used to model acoustic signals in sonar

– not generally used for audio/speech signals

RF

Front-End

Digital

“Front-End”

DetectSignal

Estimate

SignalParameters

Classify

Emitter

Compress

& Data Link

Sampling

Sub-System

DigitalAnalog ←

Receiver From Our Case Study

• Lowpass Equivalent Signal – used as a conceptual tool to aid analysis/design

– used as the actual representation in real processing

Uses of These Ideas (cont )



22/26

Uses of These Ideas (cont.)

• Analytic Signal

– generally used as a conceptual tool to prove results

– usually applied directly to the continuous-time RF bandpass signal

– There are occasions where we actually compute the analytic signal

of a real-valued digital signal,• but usually applying it some real-valued lowpass signal.

– See MATLAB Warning Below

• Hilbert Transform of a Signal – generally used as a conceptual tool to prove results

– There are occasions where we actually compute the Hilbert

transform of a real-valued digital signal,

• but usually applying it some real-valued lowpass signal.

– See MATLAB Warning Below

MATLAB Warning



23/26

MATLAB has a command that is called “hilbert”. The Help

entry on MATLAB for this command is:

“HILBERT(X) is the Hilbert transform of the real part of

vector X. The real part of the result is the original real

data; the imaginary part is the actual Hilberttransform.”

Thus, executing hilbert(x) does NOT return the Hilberttransform of x;

• It gives the analytic signal – see (S).

MATLAB Warning

Summary of Relationships



24/26

Summary of Relationships

( ) ( )cos[2 ( )]

ˆ( ) ( )sin[2 ( )]

( ) Re{ ( )}

ˆ( ) Im{ ( )}

( ) ( )cos(2 ) ( )sin(2 )

ˆ( ) ( )sin(2 ) ( )cos(2 )

******

o

o

a

a

i o q o

i o q o

x t A t f t t

x t A t f t t

x t x t

x t x t

x t x t f t x t f t

x t x t f t x t f t

π θ

π θ

π π

π π

= +

= +

− − − − −

=

=

− − − − −

= −

= +

BP Signal & Its Hilbert Transform

2 2

*************************ˆ( ) ( ) ( )

ˆ( )( ) arctan 2

( )o

A t x t x t

x t t f t

x t θ π

= +

⎧ ⎫= −⎨ ⎬

⎩ ⎭

X( f )

f o− f o f

B

t f t xt

t xt A

et At x

et xt x

t x jt xt x

oa

a

t t f ja

t f jla

a

o

o

π θ

θ π

π

2)]([)(

)()(

***********

)()(

)()(

)(ˆ)()(

)](2[

2

−∠=

=

=

=

+=

+

Signal Analytic

f o− f o f

X a( f )

B

2

( )

( ) ( )

( ) ( )

( ) ( ) ( )

*****************

( ) Re{ ( )}

( ) Im{ ( )}

( ) ( )cos[ ( )]

( ) ( )sin[ ( )]

ˆ( ) ( )cos(2 ) ( )sin(2

o j f t

l a

j t

l

l i q

i l

q l

i

q

i o o

x t x t e

x t A t e

x t x t jx t

x t x t

x t x t

x t A t t

x t A t t

x t x t f t x t f

π

θ

θ

θ

π π

−=

=

= +

=

=

− − − −

=

=

− − − −

= +

Equiv. Lowpass Model

2 2

)

ˆ( ) ( )cos(2 ) ( )sin(2 )

*****************

( ) ( ) ( ) ( )

( )( ) ( ) arctan

( )

q o o

l i q

q

l

i

t

x t x t f t x t f t

A t x t x t x t

x t t x t

x t

π π

θ

= −

= = +

⎧ ⎫⎪ ⎪= ∠ = ⎨ ⎬

⎪ ⎪⎩ ⎭

f

X l( f )

B/2 –B/2

Sampling Rate Needed for ELP Signal



25/26

Sampling Rate Needed for ELP Signal

Given a complex-valued equivalent lowpass signal, what is anappropriate sampling rate to use?

f

X l( f )

B/2 –B/2

Sampling this signal is no different than sampling some real-

valued lowpass signal: choose F s > 2 f max

…. in this case gives F s > 2( B/2) = B.

Now does this make sense?

To answer this question… look at the ELP signal’s Fourier

transform:

Sampling Rate Needed for ELP Signal (cont )



26/26

Sampling Rate Needed for ELP Signal (cont.) Now does this make sense?

Bandpass Sampling on the corresponding bandpass signal (BPS)

….would require F s > 2 B,

BUT… need only half that rate for the ELP signal!!!Do we really need only half the amount of information to

represent the ELPS as we need for the BPS?

Would that even make sense? Since ELPS ↔ BPS ?

# of Real Values for ELPS # of Real Values for BPS

It doesn’t at first!!!! BUT … the ELPS is complex

it requires a real sample value and

an imaginary sample value for each signal sample

[(I+Q) @ F s = B] = [BPS @ F s = 2B]

Bandpass Lowpass Equivalent

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