Group delay

1. Nov 14, 2007

antonantal

Why is the group delay of a filter inversely proportional to the filter bandwidth? For example in a color TV transmitter the luma signal must be delayed with respect to the chrominance signal before they are added together because the luma signal has a larger bandwidth.

2. Nov 14, 2007

stewartcs

If memory serves me correctly it is delayed in order to prevent any significant alteration of the signal shape.

It's the Fourier Transform Operation of Time Scaling.

The theorem basically says that time and bandwidth have something of an inverse relationship to one another. To transmit a signal in a shorter amount of time, more bandwidth is required, and vice versa.

Last edited: Nov 14, 2007
3. Nov 14, 2007

rbj

because if

$$x(t) = \mathcal{F}^{-1} \left\{ X(f) \right\}$$

then

$$x(at) = \mathcal{F}^{-1} \left\{ X(f/a) \right\}$$

for a > 0 .

and you can model a bandpass signal as a lowpass signal multiplied by a sinusoid. think of the lowpass signal as an "envelope" multiplying the sinusoid.

if the bandwidth gets smaller, then a gets smaller and value that t has to get to, for a t to get to the same value, must be larger. then the sense of scale of time is larger. all parameters that are based on time, including group delay or phase delay (say at a frequency of DC) must be larger.

makes good sense, but since they are yanked apart in the reciever before decoding the chroma signal and getting back to the RGB, the delay could have been dealt with at the receiver. it's a convention or protocol where they deal with it and seems most natural to put in the delay of the filter of the transmitter in with the transmitter and that of the receiver in with the receiver.

4. Nov 14, 2007

marcusl

Probably an economics decisions dating from the early days of tv. Putting one delay line in the transmitter is cheaper than 100's of millions of delay lines in the receivers.

5. Nov 15, 2007

antonantal

I don't really understand this. The property that you mentioned of the Fourier transform says that a signal with steeper variations has a wider spectrum. It relates the signal to it's bandwidth not the group delay to the signal bandwidth.

6. Nov 15, 2007

rbj

group delay (and phase delay) are really properties of an LTI system not of a signal. but since an LTI system is fully described by its impulse response (and an impulse response is a signal) sometimes we mess up semanics with words like "relates the ... group delay to the signal bandwidth" should really be "relates the ... group delay to the system bandwidth" or "relates the ... group delay to the filter bandwidth"

give me until this evening and i'll post a phase shift, phase delay, and group delay treatise. i started something like this at Wikipedia, but they have kicked me out since, and have been unable to finish it.

but the main point remains. reduce bandwidth of an LPF and you lengthen the impulse response by the reciprocal factor. that means, say you reduced the bandwidth by a factor of 4 (multiplied the bandwidth by 1/4), everything that was happening at 1 ms is now gonna happen at 4 ms. your sense of scale of time got multiplied by 4. why should some parameters of time (like what the impulse response does at 1 ms) get multiplied by 4 but other time parameters (like phase delay or group delay at DC) not get multiplied by 4? that doesn't make sense. if you shrink your ruler or tape measure down by a factor of 4 and then remeasure a bunch of objects that you had previous measured, they will all read 4 times bigger than they used to be, on your newly miniturized ruler.

Last edited: Nov 15, 2007
7. Nov 15, 2007

antonantal

I think I understand it from a point of view. One more thing: a filter with a steeper transfer characteristic needs more sections and the signal takes more time to travel through more sections, right? But a filter with the same steepness in the transfer characteristic but with a lower cutoff frequency wouldn't require more sections. So where does the delay come from?

Last edited: Nov 15, 2007