How to estimate the bandwidth of a square wave

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ZengLei

Joined: 24 Jan 2006
Posts: 102
Location: WuHan China PR

08 Mar 2008 10:39 Re: How to estimate the bandwidth of a square wave

Hi all

when i know the rising time of a square wave,how to get its bandwidth?

i see a formula 0.35/Trising=BW

is that correct?
how that formula comes?

thanks~~~~~~~~~~~

Added after 59 minutes:

The following is a relational expression between risetime of pulse waveform and the highest frequency components contained in spectrum;

Fmax = 0.35/Tr
where,
Tr = risetime of pulse (s),
Fmax = the highest frequency components (Hz).

The expression has been mentioned without any explanation in a large number of documents as if it is a matter of course. I do not think its basis is self-explanatory at all.
I think you can see intuitively that waveforms of speedy risetime have high frequency components. However,

1) the reason why Fmax and Tr are in an inverse proportion, and
2) what is the basis of the constant "0.35,"

cannot be matters of course by any means. What kind of basis on earth, could this expression be derived from?

FvM

Joined: 22 Jan 2008
Posts: 5161
Helped: 767
Location: Bochum, Germany

08 Mar 2008 11:56 Re: How to estimate the bandwidth of a square wave

As any estimation, the said rule-of-thumb depends on assumptions and has limited accuracy. For a mathematical exact analysis, you may consider fourier series of a periodical trapeziodal pulse shape. It has obviously infinite bandwith, but you can calculate fmax for a given percentage of energy.

If you assume frequency characteristics as second-order bessel or gaussian filter. You get rise times (10%-90%) of the said magnitude with fmax as cut-off frequency.

Kral

Joined: 28 Mar 2005
Posts: 1182
Helped: 187

08 Mar 2008 20:37 Re: How to estimate the bandwidth of a square wave

ZengLei,
The formula BW = .35 / Trise is based on the following assumptions:
.
Rise time is defined as the time required for the signal to go from 10% to 90% of its ultimate change.
.
The load on the circuit is a simple lag. In other words, the output is taken downstream of the internal series resistance, and in parallel with a capacitor.
.
Based on these assumptions, the formula can be derived analytically. Let me know if you want the derivation.
Regards,
Kral

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ZengLei

Joined: 24 Jan 2006
Posts: 102
Location: WuHan China PR

09 Mar 2008 7:51 How to estimate the bandwidth of a square wave

OK,Kral,would u pls give the detailed derivation of the formula.

thanks~~~

FvM

Joined: 22 Jan 2008
Posts: 5161
Helped: 767
Location: Bochum, Germany

09 Mar 2008 11:07 Re: How to estimate the bandwidth of a square wave

(ln(0.9)-ln(0.1))/(2•pi)

IanP

Joined: 05 Oct 2004
Posts: 6490
Helped: 1542
Location: West Coast

09 Mar 2008 11:56 Re: How to estimate the bandwidth of a square wave

if you know the rising time of a "square wave" it's not a square wave any more since the harmonics spread out to infinity .. and that it's bandwidth ..

just playing devil's advocate ..

cheers,
ianp

ZengLei

Joined: 24 Jan 2006
Posts: 102
Location: WuHan China PR

09 Mar 2008 12:44 How to estimate the bandwidth of a square wave

FvM

:
Vin=Uo
Vout=Uo[1-exp(-t/RC)]

T90%=-RCln0.1
T10%=-RCln0.9

Trise=T90%-T10%=RC(ln0.9-ln0.1)=RCln9

1/(2*pi*RC)=K/(RCln9) ==> K=ln9/(2*pi)=0.35

i'm i right?

We can use this formula to estimate the Bandwidth of a one-pole sysytem.

But can we use this formula to estimate the Bandwidth of a square waveform??

Thanks

FvM

Joined: 22 Jan 2008
Posts: 5161
Helped: 767
Location: Bochum, Germany

09 Mar 2008 13:39 Re: How to estimate the bandwidth of a square wave

It has all been said! The bandwith estimation is under the assumption of first order system (as I said, second order is basically the same order of magnitude). A square wave has infinite bandwith, there is nothing left to calculate. An ideal trapezoidal waveform has also infinite bandwith. You can read the fourier series from any mathematical handbook.

Kral

Joined: 28 Mar 2005
Posts: 1182
Helped: 187

11 Mar 2008 21:59 Re: How to estimate the bandwidth of a square wave

ZengLei,
I didn’t follow your fifth equation, so here’s my development:
Assume the input and output are initially at zero, V is the instantaneous output voltage, Vf is the final output voltage, T is the RC time constant. Then
1) V = Vf[1-exp(-t/T)
2) V = Vf –Vfexp(-t/T)
3) V10 = .1Vf
4) .1Vf-Vf = -Vfexp(-t10/T)
5) .9 = -exp(-t10/T)
6) ln(.9) = -t10/T
7) t10=-Tln(.9)
Cool

Similarly, t90 = Tln(.1)
9) trise = t90 - t10 = [-Tln(.1) + Tln(.9)]
10) T =1/w = 1/(2 pi f) for a 1st order lag
11) From 9), 10, trise = [-ln(.1) + ln(.9)] / (2 pi f)
12) trise = .35/f
Regards,
Kral