In article , Paul Keinanen
writes:
On 05 Feb 2004 17:35:31 GMT, (Avery Fineman)
wrote:
Signal to noise ratio changes as the _square_root_ of bandwidth
change.
I do not quite understand this.
Usually the signal to noise ratio is defined as the signal power S
compared to the noise N (or compared to S+N). In a white noise
environment with constant noise density, the noise power is directly
proportional to bandwidth.
If the noise bandwidth is larger than the required signal bandwidth,
reducing the noise bandwidth will not affect the signal but only
reduce the noise power directly proportional to the bandwidth
(dropping the bandwidth to 1/4 will drop the noise power by 1/4 or 6
dB and increase the SNR by 6 dB).
However, if you looking at the noise _voltage_, it will drop by the
square_root of the bandwidth. But dropping the bandwidth to 1/4 will
drop the noise voltage to 1/2, which is again 6 dB.
My apologies for not stating a reference. Was thinking in terms of
voltage, a bad habit picked up with vacuum-state electronics many
moons ago. I've been working with FETs lately and am too
(for shame) conditioned to thinking in terms of signal voltages.
In the OP's case, some of the signal sidebands are also cut, thus also
reducing the signal power. Are you assuming something about the
spectral or phase distribution of these sidebands (e.g. adding
coherent side bands produce the sum of the sideband _voltages_, while
adding noise from two side bands with random phase only produces a sum
of sideband _power_).
Random noise POWER always adds directly. If the spectral
distribution of noise power is uniform, the noise power through a
bandwidth-limiting device is proportional to the bandwidth of that
device. Ergo, if there is a change in bandwidth, the amount of
noise power change is directly proportional to bandwidth change.
Normally for such things as radar, the receiver bandwidth is kept
wide to preserve the echo's shape (a very rectangular pulse if the
target is very reflective and a plane (not airplane) surface. The
leading edge sharpness is very useful in determining precise range
to the radar target.
In other applications, such as limiting the pulsed signal bandwidth
to just a few sidebands on either side of the carrier, the received
pulse is rounded, more rounded as the sidebands get down to just
one set closest to carrier. If just a portion of the first (or major) SB
set, the shape resembles the "cosine-squared" envelope, nowhere
close to rectangular. That is sometimes called a "matched filter"
condition where the bandwidth is equal to the pulse duration time
inverse.
At RCA EASD, there was an R&D program for aircraft anti-collision
that involved many carrier frequencies all spaced at 1 MHz intervals
with 1 uSec pulse widths. Deliberate receiver limiting was used so
the discrimination between frequencies depened highly on the
matched-filter distortion. At 1 MHz away, a matched-filter would
pass only the sideband content of the first two sidebands on one
side; the resulting RF envelope would have a rounded "bow tie"
shape with a reduced peak envelope. That made it relatively easy
to discriminate against adjacent channels. While predicted in
theory by the system designer, it nonetheless had a number of the
senior staff (all very used to conventional RF pulse techniques)
scratching their heads and doing many simulations. It worked very
well in hardware but I've not seen any similar explanation in any
texts before or since.
Something vaguely similar goes on in digital TV now, the video
information sent digitally and the resulting digital pulse train shaped
to reduce sideband content. That allows a nearly one-third bandwidth
reduction so that a 6 MHz wide TV channel allotment can accept the
information requiring at least 16 MHz bandwidth under conventional
video modulation techniques. Obviously there is also the MPEG
video compression also a part of the bandwidth reduction. Digital TV
has an almost infinite S:N ratio down to the lower limit of its processing
capability, none of the noise "snow" effects seen with early days of
analog TV and distant TV stations "down in the mud."
Remotely, vaguely similar (:-) is the now-conventional 56K telephone
line modem capable of sending high digital data rates over a 3 KHz
(or less) telephone line bandwidth. While most of that bandwidth
reduction is due to using a combination of AM and PM, there exists
bandpass filtering of one sort or another to limit the sideband content
of that combinatorial modulation. All of the techniques mentioned
work in concert to present the best signal-to-noise ratio of each system.
Ever since the RCA SECANT project, I've gotten away from the
conventional wide-bandwidth radar-like thinking in terms of RF pulses.
Sometimes pulse shape distortion can be used to advantage. It
depends on the application and not "staying in the box" all the time
(holding to conventional thinking).
My apologies again for not stating my references to voltage instead of
power in the previous message.
Len Anderson
retired (from regular hours) electronic engineer person