Analog raised cosine filter ?

[Georgi]

Hi,
I was wondering if there is a way to implement an analog Pulse shaping raised cosine filter ?
I looked in the internet, but only found digital implementations.

Georgi

[chris_leyson]

Hi Geogi, in principle I don't see why not, I found this on ebay, http://www.ebay.co.uk/itm/3-X-1MHZ-RAISED-COSINE-FILTER-/111992476524

[T3sl4co1l]

Well, technically you wouldn't, because all-pole analog filters have polynomial transfer functions.  Cosine is a transcendental function, so has no equal in finite polynomials (i.e., filters without infinite numbers of L and C).  The best you can do is an approximation.

The roll-off region (say 0 to 6dB insertion loss) should be pretty easy to reproduce, but the asymptotes cannot simply go to zero; you're left with a residual level in the cutoff band.

An approximation could be specified by the sharpness of the cosine (beta in these https://en.wikipedia.org/wiki/Raised-cosine_filter formulas) and desired stopband attenuation (i.e., it doesn't go to zero in the "otherwise" region, but is at least some amount of attenuation there).

To match the most desirable property (periodic zeroes in the impulse response), you'd probably want to use a different approach, though.  I don't know enough about real analysis to do that (but who cares, anyway; just have a computer solve it).

That said, at least if your cutoff frequency is conveniently high: you can use a transmission line, cut so that its impedance varies as a function of length, to implement the impulse response.  The total length of the line corresponds to the length of the impulse response, and needs to be N/2 times the wavelength for N zeroes.  Actual stopband performance will depend on how precisely cut the transmission line is, and its losses.  (It's not impossible to get infinite attenuation in the stopband, but you need the reflected wave to cancel perfectly with the incident wave, which isn't going to happen for much frequency range, because transmission line losses rise as ~sqrt(f).)

Probably, SAW filters implement this pretty regularly, at more modest frequencies.  But as those are rather special purpose items, you'd probably never see it, except for a few you might randomly wander across, made for a special frequency.

Tim

[uncle_bob]

Hi,
I was wondering if there is a way to implement an analog Pulse shaping raised cosine filter ?
I looked in the internet, but only found digital implementations.

Georgi

Hi

A lot depends on what you are trying to do. Are you trying to shape a generated pulse or filter a received pulse? As mentioned above, you can only do it just so well and then things fall apart. For the pulse generation side, there are a few odd non-linear things you can throw into the mix.

Bob

[German_EE]

For pulse shaping I would use a network of series resistors with capacitors to ground forming an RC low pass filter. I use one of these for shaping the keying waveform in a CW transmitter, a rising square wave pulse going in produces a beautiful rising raised cosine at the output. Values depend on your required rise time and are computed using the formula risetime= 1/(2 PI R C)

[coppice]

Are you sure you want a raise cosine filter, and not a root raised cosine filter?

You can make reasonable approximations to raised cosine and root raised cosine by analogue means. You can also make adaptive forms, to achieve dynamic equalisation in conjunction with the root raised cosine. These were implemented with discrete parts in the 70s, and by switched capacitors in the 80s. Numerical methods pretty much took over after that.

[T3sl4co1l]

For pulse shaping I would use a network of series resistors with capacitors to ground forming an RC low pass filter. I use one of these for shaping the keying waveform in a CW transmitter, a rising square wave pulse going in produces a beautiful rising raised cosine at the output. Values depend on your required rise time and are computed using the formula risetime= 1/(2 PI R C)

Mmmm, no... not quite.

Also, that's time domain, not frequency domain.

Time domain (impulse) response of lumped constant filters can only be a series of the form t^N exp(-t/tau), where tau can be complex (in which case there will always be a term with its complex conjugate as well).  Usually, as you cascade filter sections, the N's go up.  With RC filters, tau is always real, which is to say, the filter has a low Q.

A Bessel filter could be massaged into giving a reasonable time-domain approximation of a "raised cosine" function.  The leading edge is causal, so you've got that in the bag.  The settling time, however, goes on forever (a series of exponential tails), so cannot be made exactly equivalent.  This is precisely the same behavior, and reason, where the frequency response cannot be exactly matched in the asymptotic cutoff range.

The Fourier transform of a TDRC (time domain raised cosine) is an oscillating function of frequency, so you would have periodic zeroes where there is no gain at harmonic frequencies, and asymptotic cutoff for non-harmonic frequencies.  And, for the same reasons, it could be better approximated with a transmission line component (where the transmission line cross section is inverse, in a sense, from the FDRC case).

Tim

[Georgi]

Thanks for the replies guys, learned a lot from them.

I didn't think my question would raise such a discussion.
Anyway I know normally I would want to use a root raised cosine, and not a raised cosine pulse shaping filter.
I think it was due to the multiplication at transmitter and receiver that we get squared raised cosine, which has doesn't meet the
Nyquist rule(not the sampling theorem, but the one that has to do with the orthogonality of the pulses).
And that is why a root raised cosine filter is used.
I asked this question in the first place, because i have a colleague of mine, that wants to use Chebyshev filter as a pulse
shaping filter. Since i have just finished university I am still not 100% sure in my knowledge, but I think it is a bad idea to use Chebyshev as a Pulse shaping filter, because of the Ripples in the Transfer function and the non linear phase response. That is why i suggested raised cosine filter, but my colleague said it couldn't be done analog, and i though I'll ask here.
Georgi

[T3sl4co1l]

A Bessel is the best, simplest synthesized analog filter for pulse applications; it exhibits minimum group delay.

There are also filters designed for minimal over/undershoot and settling time (which I don't think are analytically synthesized, but optimized numerically instead?).

Neither will give you the orthogonality condition on sampling, but they can give you better continuous-time filtering (i.e., instead of humps above and below zero, with zeroes at the sample rate, the response falls off as quickly as possible for t > tau).

You are correct that Chebyshev is the worst option; it has rapid frequency attenuation but very poor pulse characteristics.

Tim

[Georgi]

Neither will give you the orthogonality condition on sampling, but they can give you better continuous-time filtering (i.e., instead of humps above and below zero, with zeroes at the sample rate, the response falls off as quickly as possible for t > tau).

That is what I meant by mentioning orthogonality. On sampling all the neighbor pulses are at zero, and the effects of ISI are minimized, or am I wrong ?

[coppice]

You are correct that Chebyshev is the worst option; it has rapid frequency attenuation but very poor pulse characteristics.

If you can't get worse than Chebyshev you aren't really trying.

There's no magic in filters. If you push harder in one direction, things get worse in another. Anything achieving a fast roll off without a large number of poles and/or zeros is going to do funky things in phase. That's not useful for pulse shaping.