Why fir filters have properties of linear phase?

[rednewguy]

linear phase

i read in a book that fir filters possess the property of linear phase. what does that mean.
thanks

 

[electronics_kumar]

linear phase

phase shift offered by that filter is a constant ..it will vary for different frequencies

 

[Kral]

Re: linear phase

rednewguy,
Linear phase means that in the passband, the phase shift is directly proportional to frequency, i.e., the phase shift varies linearly with frequency. Phase linearity is important for preserving the waveforms of complex (non-sinusoidal) signals. For example, consider a waveform consisting of a fundamental and a third harmonic. If the fundamental is shifted bye -10 radians, and the third harmonic is shifted by -30 radians, the output will be a delayed version of the input. The output will be delayed by an amount equal to 10/wo, where wo is the radian frequency of the fundamental. The waveshape of the output will be identical to the waveshape of the input.
~
FIR filters can have the linear phase property. A necessary and sufficient condition for phase linearity is the the summing coefficients be symmetrical about the center coefficient. For example, if the coefficients are c0, c1, c2, c3, c4, then if co = c4 and c1 = c3, then the filter will be phase linear. If this symmetry condition is not met, then the fir filter will not be phase linear.
Regards,
Kral

 

[rednewguy]

Re: linear phase

hi kral
why shud the third harmonic be shifted by -30 radians. why not the same as fundamental(-10 radians)

 

[Kral]

Re: linear phase

rednewguy,
This is probably best explained by an example. To simplify the arithmetic, I'll use hertz instead of radians/second in this example. Also, to make most of the numeric results come out even, I'll use an input waveform consisting of a fundamental and a second harmonic.
Let fo = 10 Hz (Period = 0.1 Seconds
Let f2 = 20 Hz (Period = 0.05 Seconds)
~
Let's assume the the phase shift of the fundamental is 10 degrees. This corresponds to a delay of 0.1Seconds * (10/360) Seconds = .00277777 Seconds.
~
If the filter is linear phase, then the delay of the 2nd harmonic must = the delay of the fundamental. 0.05 Seconds * (20/360) Seconds = = .00277777 Seconds.
~
The thing to remember is that the delay that corresponds to a given phase shift is inversely proportional to the frequency.
Regards,
Kral

 

[mimomod]

Re: linear phase

Hi guys,

Kral has given very simple, yet very good explanation on the linear phase.

I just want to add some further information. Hopefully it can give some light.

The linear phase property is very important for a filter. For some applications, we need a filter which has an output signal which is merely a delay and amplitude-scaled version of the input signal.

For example, in designing an amplifier. In this case we need a filter which is able to amplify the signal input without altering it. As Kral has explained it, a signal theoretically may be decomposed into many harmonics, i.e. a sinusoids with different frequencies which are multiple of a basic frequency (remember Fourier series or Fourier transform).

If we still have the same harmonics but we shift some of them relative to the others, we will get different waveform. Thus, the original signal is distorted.

The linear phase property preserves the relative positions of those harmonics by shift all harmonics with the same delay in time. This same delay in time is transformed into different phase shift for each harmonic, i.e. the phase shift is linearly dependent on the frequency of each harmonic. By the way, this same shift for all harmonics is refered to as group (envelop) delay of the filter.

Mathematically, the group delay of the filter is computed by doing derivative of the phase response with respect to frequency. And we all know that if the phase response is linear, then the derivative of this linear function will be a constant.

best
mimomod

 

[elexhobby]

Re: linear phase

As said above, group delay in general is given by the negative derivative of phase wrt frequency.. Can somebody tell me how to prove this general result.. I couldn't find the proof in most of the books I've referred.. The linear phase delay is the only formula whose proof is supplied, & then the general formula is presented without proof, saying that the linear phase delay is just a special case of the general phase delay formula..
Thanks in advance,

 

[Lupin]

Group Delay-Transit Time-Measurement-Average Group Delay

Excuse me guys,

I have some question for u...

1- In a given RF system , what's the difference between the group delay and the transit time?

In other words, what's the physical concept of the group delay? is it the time spent by the signal to go through a component / medium?

2- Given an RF device, how can the Group delay be measured?
Can we measure the Phase envelope vs Frequency and then post process the data making a neagative derivate? I would need a cofnirmation on this.

3- Then, if I know the Group Delay of every component belonging to a system, the Overall Group delay is given by the average group delay of all the chain?

In the Satellite field, I realize it's very important.
Hope to have a fruitful discussion with you.

Regards

Lupin

 

[electronics_kumar]

linear phase

THEN ......what is the differeence between group delay and phase delay?...

 

[Kral]

Re: linear phase

electronics_kumar,
Phase delay is phase divided by frequency at a particular frequency.
Group delay is the slope of the phase vs frequency curve.
If the device has linear phase, the group delay and phase delay are equal.
Regards,
Kral

 

[electronics_kumar]

linear phase

Kral
SOMEWHERE i found that product of phase and group delay is square of velocity of sound

 

[Kral]

Re: linear phase

electronics_kumar,
This doesn't look right, since the units are inconsistent:
~
Phase is dimensionless
Group delay has the unit of time
Speed^2 has the unit (distance^2/time^2).
~
Velocity of sound can be expressed as
V = d/Ψ, where
. d = distance traversed by a sound wave
. ψ = group delay
~
Regards,
Kral

 

[johnseven]

Re: linear phase

electronics_kumar,
This doesn't look right, since the units are inconsistent:
~
Phase is dimensionless
Group delay has the unit of time
Speed^2 has the unit (distance^2/time^2).
~
Velocity of sound can be expressed as
V = d/Ψ, where
. d = distance traversed by a sound wave
. ψ = group delay
~
Regards,
Kral

Guys, really cool discussion.
Refreshed my rusty knowledge of the subject.
Regards
J7

 

[gunsuresh]

Re: linear phase

That means the phase shift offered by that filter is changing in linear steps.

 

[Aryanfar]

Re: linear phase

i read in a book that fir filters possess the property of linear phase. what does that mean.
thanks

as u know each filter has wo characteristic one is related to it magnitude and another is about its phase because the frequency responce is in general a complex function and has magnitude and phase. the magnitude detemines the amplification or attenuation of each freq. and the phse is yje phase deifference caused by filter. phase difference in time domain is related to delay and if filter have nonlinear phase then each freq experince different dalay and distortion is occurred which is called phase distortion if phase is linear then this delay for all freq would be constant and no phase distortion will exist. fir filters if have somekind of symmetry in iets imulse response the its phase would be linear.