Definition of Group (Envelope) Delay

I've been racking my brain on this one the last couple of days.  So, group
delay is defined as the derivative of phase with respect to frequency.

My simple question is: why?

We would all agree that the time delay of a sinusoid induced by a change in
phase (phase delay) is equal to the ratio of the change in phase to the
frequency.

Also, for a linear phase response, the derivative would be equal to the
ratio.  Both phase delay and group delay would be constant.

I have been searching for a while now and haven't found anything beyond 'it
can be shown that...'  In Richard Lyons' book, he simply uses 'is defined
as...'

Could someone provide, or refer me to, a derivation showing the
relationship between an amplitude modulated sinusoid and group delay?

Cheers,
-Dan

"dszabo" <62466@dsprelated> writes:

> I've been racking my brain on this one the last couple of days.  So, group
> delay is defined as the derivative of phase with respect to frequency.
>
> My simple question is: why?
>
> We would all agree that the time delay of a sinusoid induced by a change in
> phase (phase delay) is equal to the ratio of the change in phase to the
> frequency.
>
> Also, for a linear phase response, the derivative would be equal to the
> ratio.  Both phase delay and group delay would be constant.
>
> I have been searching for a while now and haven't found anything beyond 'it
> can be shown that...'  In Richard Lyons' book, he simply uses 'is defined
> as...'
>
> Could someone provide, or refer me to, a derivation showing the
> relationship between an amplitude modulated sinusoid and group delay?
>
> Cheers,
> -Dan

Dan,

First of all, group delay is characteristic of a system, not a signal.
Thus you wouldn't measure the group delay of "an amplitude modulated
sinusoid."

Put simply, group delay is a measure of the time through the system as a
function of frequency. If it is a linear-phase system, the delay through
the system is constant and so is the group delay.
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On 11/26/12 5:27 PM, dszabo wrote:
> I've been racking my brain on this one the last couple of days.  So, group
> delay is defined as the derivative of phase with respect to frequency.
>
> My simple question is: why?
>

maybe take a look at:

   http://en.wikipedia.org/wiki/Group_delay_and_phase_delay

it sets up the problem, but doesn't do the math to prove the result (but 
i think it provides all of the conditions so that it's a matter of 
cranking through the math).

if transfer function, H(s), is evaluated at s = j omega

     H(j omega)  =  |H(j omega)| e^(j phi(omega) )

and the input signal is

     x(t)  =  A(t) cos(omega t)

and A(t) varies slowly enough to be deemed an "envelope", then the 
output is virtually

     y(t)  =   |H(j omega)| A(t - T_g) cos( omega (t - t_phi) )

the *envelope* is delayed by T_g which approaches in the limit

    T_g =  -phi'(omega)

> We would all agree that the time delay of a sinusoid induced by a change in
> phase (phase delay) is equal to the ratio of the change in phase to the
> frequency.

well, *phase* delay is

   T_phi  =  -phi(omega)/omega

just crank through the math.  apply limits where appropriate.

> Also, for a linear phase response, the derivative would be equal to the
> ratio.  Both phase delay and group delay would be constant.

yes, that's the case.  and they gotta be the same constant.

in fact, even for non-linear phase, it's still the case that the phase 
delay and group delay are equal to the same value at DC.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Thanks for the reply!

This wikipedia article did a nice job of summarizing the information I
found from google: 
http://en.wikipedia.org/wiki/Group_delay_and_phase_delay

You'll see that for AM signals convolved with a tranfer finction, h(t),
that the amplitude, A(t), and the carrier, cos(omega*t+theta) are time
shifted by tao-sub-g (the group delay) and tau-sub-phi (the phase delay).

It then goes on to claim that it can be shown that these values are equal
to the inverse of the derivative of the phase with respect to frequency,
and the ratio of the phase with respect to frequency, respectively.

I would like to see the derivation that proves that the amplitude signal
would be time shifted by the derivative of the phase with respect to
frequency.

Cheers,
-Dan

>if transfer function, H(s), is evaluated at s = j omega
>
>     H(j omega)  =  |H(j omega)| e^(j phi(omega) )
>
>and the input signal is
>
>     x(t)  =  A(t) cos(omega t)
>
>and A(t) varies slowly enough to be deemed an "envelope", then the 
>output is virtually
>
>     y(t)  =   |H(j omega)| A(t - T_g) cos( omega (t - t_phi) )
>
>the *envelope* is delayed by T_g which approaches in the limit
>
>    T_g =  -phi'(omega)


Now we're getting to the heart of it!

OK, so I'm with you for evaluating H at j omega.  And I know that there is
a laplace transform relationship between the e^(s*tao) and a dirac delta
shifted in time.

SO, A multiply in the s-domain would correspond to a convolution in time
domain, which would shift the input signal by tao.  HOWEVER, this does not
explain why my envelope and carrier functions get shifted by different
amounts.

Thoughts?

Cheers,
-Dan

dszabo <62466@dsprelated> wrote:
> I've been racking my brain on this one the last couple of days.  So, group
> delay is defined as the derivative of phase with respect to frequency.
 
> My simple question is: why?
 
> We would all agree that the time delay of a sinusoid induced by a change in
> phase (phase delay) is equal to the ratio of the change in phase to the
> frequency.

I first learned it in terms of phase and group velocity, where phase
velocity is w/k, and group velocity dw/dk.

(w, omega is angular frequency, usually in radians per second, and k
is the wave vector or wave number, in, usually, inverse meters.)

That is, the wave equation will have terms in (kx-wt) and (kx+wt). 

In a system that is close to linear, such as light through a 
dielectric, phase velocity is the velocity of phases (crests
and valleys) in the EM wave, group velocity of changes in the
amplitude, or envelope. 

If you send information by modulating the wave, such as AM on
a light wave carrier, the information travels at the group
velocity.

Now, divide all that by l (length) and you get phase and group delay.

> Also, for a linear phase response, the derivative would be equal to the
> ratio.  Both phase delay and group delay would be constant.

Group delay works when it is linear, or close to linear. That is,
when the second derivative is small such that it can be ignored
in the Taylor series expansion. Near resonance, that isn't true,
especially at places where the group velocity (and delay) go
negative.
 
> I have been searching for a while now and haven't found anything beyond 'it
> can be shown that...'  In Richard Lyons' book, he simply uses 'is defined
> as...'
 
> Could someone provide, or refer me to, a derivation showing the
> relationship between an amplitude modulated sinusoid and group delay?

Group and phase velocity should be in some not too far from introductory
physics books. They work well for many dielectrics which are just close
enough to linear over the interesting range of frequencies.

-- glen

On 11/26/12 6:50 PM, dszabo wrote:
>> if transfer function, H(s), is evaluated at s = j omega
>>
>>      H(j omega)  =  |H(j omega)| e^(j phi(omega) )
>>
>> and the input signal is
>>
>>      x(t)  =  A(t) cos(omega t)
>>
>> and A(t) varies slowly enough to be deemed an "envelope", then the
>> output is virtually
>>
>>      y(t)  =   |H(j omega)| A(t - T_g) cos( omega (t - t_phi) )
>>
>> the *envelope* is delayed by T_g which approaches in the limit
>>
>>     T_g =  -phi'(omega)
>
>
> Now we're getting to the heart of it!
>
> OK, so I'm with you for evaluating H at j omega.  And I know that there is
> a laplace transform relationship between the e^(s*tao) and a dirac delta
> shifted in time.
>
> SO, A multiply in the s-domain would correspond to a convolution in time
> domain, which would shift the input signal by tao.  HOWEVER, this does not
> explain why my envelope and carrier functions get shifted by different
> amounts.
>
> Thoughts?

i'm gonna change the notation a little.  but not much.

this is a special input signal, not a totally general signal.  you have 
a bandlimited (but otherwise general) envelope, a(t), and it is 
modulating a sinusoid resulting in input

    x(t)  =  a(t) cos(w0 t)

          =  a(t)  1/2 ( e^(j w0 t) + e^(-j w0 t) )

the phase of the input sinusoid (or "co"sinusoid) can be set to zero by 
how t is defined.

the fourier transform (using angular frequency) of the above is

     X(jw)  =  1/2 ( A(j(w+w0)) + A(j(w-w0)) )

where A(jw) is the F.T. of a(t).

if H(s) is the transfer function, the F.T. of the output is

     Y(jw)  =  H(jw) X(jw)

the transfer function H(jw) is, in polar form

     H(jw)  =  |H(jw)| e^(j phi(w))

because a(t) is slowly varying (it's an envelope), meaning *bandlimited* 
to much less than w0.  that means that

     A(jw) =  0    except when  |w| << w0

now here is where the approximation is made for H(jw) at frequencies w 
that are in the neighborhood of w0, that is |w-w0| << w0 :

     H(jw)  =  |H(j w0)|  e^( j (phi(w0) + phi'(w0)(w-w0)) )

and similarly for frequencies around -w0, or |w+w0| << w0 .

phi'(w0) is the derivative of phi(w) evaluated at w = w0.

now Dan, can you take it from there (that is figger out what y(t), the 
inverse F.T. of Y(jw), is) ?

the output y(t) you should be looking for is

    y(t)  =  |H(j w0)| a(t - T_g(w0))  cos(w0 (t - T_phi(w0)))

where

    T_phi(w)  =  -phi(w)/w

and

    T_g(w)  =   -phi'(w)   or  (d/dw)phi(w)


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Hmm,

I've started working through the derivation, and hope to finish it on my
lunch break.  It looks like the two side bands (A(j(w+w0)) and A(j(w-w0)))
are going to have a phase shift in common, and a phase shift difference. 
These will result in the different group/phase delays.

I would also assume that h(t) has to be real, meaning that:

|H(jw0)| = |H(-jw0)|

phi(-w0) = -phi(w0)

phi'(-w0) = phi'(w0)

Thanks so much for the help.

Cheers,
-Dan

Okay, here goes.

First, the approximation of the transfer function would be

    H(j(-w0)) = |H(jw0)| e^j(-phi(w0) + phi'(w0)(w+w0))

This is from the assumptions I made in my previous post.

Y(jw) has two non-zero regions, near +w0 and -w0.

Near +w0:
    A(j(w+w0)) = 0

Near -w0:
    A(j(w-w0)) = 0

So:

    Y(jw) = |H(jw0)|*(1/2)*( (e^j(phi(w0) + phi'(w0)(w-w0))) * A(j(w-w0))
)
                          *( (e^j(-phi(w0)+ phi'(w0)(w+w0))) * A(j(w+w0))
)

The phi'() terms cause an equal phase shift in the A() about +w0 and -w0,
respectively.  This is equivalent to shifting the orginial spectrum of A
before modulation by -phi'(w0).

Lets call this new group delay signal b(t)

    b(t)  = A(t-tao_g)
    B(jw) = A(jw)e^j(-(-phi'(w0))*w)

Plugging back into Y(jw):

    Y(jw) = |H(jw0)|*(1/2)*( ( (e^j(phi(w0)))  * B(j(w-w0)) )
                           + ( (e^j(-phi(w0))) * B(j(w+w0)) ) )

From the relationship used eariler:

    y(t) = |H(jw0)|*b(t)*(1/2)*( (e^j(phi(w0)))  * (e^j*w0*t)
                               + (e^j(-phi(w0))) * (e^-j*w0*t) )

    y(t) = |H(jw0)|*b(t)*(1/2)*( e^ j(w0*t+phi(w0)))
                               + e^-j(w0*t+phi(w0))) )

Or,

    y(t) = |H(jw0)|*b(t)*cos(w0*t + phi(w0))

Finally, plug in a() and rearrange to introduce the phase delay term:

    y(t) = |H(jw0)| * a(t-tao_g) * cos(w0(t-tao_phi))

Where,
    tao_g   = -d/dw phi(w0)
    tao_phi = -phi(w0)/w0


Thanks for all the help Robert!

Cheers,
-Dan

On 11/27/12 3:45 PM, dszabo wrote:
> Okay, here goes.
>
> First, the approximation of the transfer function would be
>
>      H(j(-w0)) = |H(jw0)| e^j(-phi(w0) + phi'(w0)(w+w0))
>
> This is from the assumptions I made in my previous post.
>
> Y(jw) has two non-zero regions, near +w0 and -w0.
>
> Near +w0:
>      A(j(w+w0)) = 0
>
> Near -w0:
>      A(j(w-w0)) = 0
>
> So:
>
>  Y(jw) = |H(jw0)|*(1/2)*( (e^j(phi(w0) + phi'(w0)(w-w0))) * A(j(w-w0)) )
>                        *( (e^j(-phi(w0)+ phi'(w0)(w+w0))) * A(j(w+w0)) )
>
> The phi'() terms cause an equal phase shift in the A() about +w0 and -w0,
> respectively.  This is equivalent to shifting the original spectrum of A
> before modulation by -phi'(w0).
>
> Lets call this new group delay signal b(t)
>
>      b(t)  = A(t-tao_g)
>      B(jw) = A(jw)e^j(-(-phi'(w0))*w)
>
> Plugging back into Y(jw):
>
>      Y(jw) = |H(jw0)|*(1/2)*( ( (e^j(phi(w0)))  * B(j(w-w0)) )
>                             + ( (e^j(-phi(w0))) * B(j(w+w0)) ) )
>
>  From the relationship used eariler:
>
>      y(t) = |H(jw0)|*b(t)*(1/2)*( (e^j(phi(w0)))  * (e^j*w0*t)
>                                 + (e^j(-phi(w0))) * (e^-j*w0*t) )
>
>      y(t) = |H(jw0)|*b(t)*(1/2)*( e^ j(w0*t+phi(w0)))
>                                 + e^-j(w0*t+phi(w0))) )
>
> Or,
>
>      y(t) = |H(jw0)|*b(t)*cos(w0*t + phi(w0))
>
> Finally, plug in a() and rearrange to introduce the phase delay term:
>
>      y(t) = |H(jw0)| * a(t-tao_g) * cos(w0(t-tao_phi))
>
> Where,
>      tao_g   = -d/dw phi(w0)
>      tao_phi = -phi(w0)/w0
>
>
> Thanks for all the help Robert!

you're welcome.

guess what you could do to spread the joy:  you could fix up that page 
at Wikipedia (anyone can edit, even anonymously) and add the rest of the 
derivation.  i don't have the time to do it now and also i have been 
officially banned from editing wikipedia (hasn't stopped me from editing 
anonymously).

anyway, just a thought.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."