# Frequency Division Multiplexing. FM SSBSC system. Bandpass Filter

1. Aug 22, 2013

### Fisher92

1. The problem statement, all variables and given/known data
Due to its bandwidth and power efficiency, SSBSC is commonly used in Frequency Division Multiplexing (FDM) systems. FDM systems are used to combine a number of separate information channels into a single transmission. Each channel uses a separate carrier frequency and the carriers are spaced evenly in the available bandwidth.

Design an FDM system to suit the following specifications.
• Available bandwidth range 100 to 120 kHz
• Number of separate information channels 4
• Information bandwidth 100 Hz to 4.5 kHz.
• Sideband suppression required in each channel is 40 dB.

In the design you need to consider the following:
• What is the frequency of local oscillator required at each channel?
• What are the parameters (Q, fc) for bandpass filter required at each channel?
• Provide a diagram of the FDM transmitter. As a minimum, this need to include blocks for the LO, modulator (what type should you use), bandpass filter and amplifiers. Think about what you can use to combine different signals for before transmission.

2. Relevant equations
NA/// useful & succinct : http://en.wikipedia.org/wiki/Frequency-division_multiplexing

3. The attempt at a solution
What is the frequency of local oscillator required at each channel?[/B]

Frequency division multiplexing (FDM) means that the total bandwidth available to the system is divided into a series of nonoverlapping frequency sub-bands that are then assigned to each communicating source and user pair.

I have a 20kHz window and 4.4kHz information signal. So I want:
Carrier frequencies at:
Channel 1: 102.2kHz
Channel 2: 107.2kHz
Channel 3: 112.2kHz
Channel 4: 117.2kHz
?

What are the parameters (Q, fc) for bandpass filter required at each channel?
This is essentially generating the SSBSC signal, as i read it?
To suppress the LSB I want a band pass centered in between the carrier of each channel + 1/2 of 2.2kHz? This would five my critical frequencys for the Bandpass filter at fc and fc+2.2kHz?

(Bandpass filter has 2 critical frequencys that dictate the center frequency.)

channel 1:
$$f_0=\sqrt(f_c1*f_c2)=\sqrt(102.2k*104.4k)=103.294kHz$$
$$Q=\frac{f_0}{BW}=\frac{103.294k}{2.2k}=46.952$$

channel 2:
$$f_0=\sqrt(f_c1*f_c2)=\sqrt(107.2k*109.4k)=108.294kHz$$
$$Q=\frac{f_0}{BW}=\frac{108.294k}{2.2k}=49.225$$

channel 3:
$$f_0=\sqrt(f_c1*f_c2)=\sqrt(112.2k*114.4k)=113.295kHz$$
$$Q=\frac{f_0}{BW}=\frac{113.295k}{2.2k}=51.50$$

channel 4:
$$f_0=\sqrt(f_c1*f_c2)=\sqrt(117.2k*119.4k)=118.295kHz$$
$$Q=\frac{f_0}{BW}=\frac{118.295k}{2.2k}=53.77$$

I want a 40dB suppression outside of the critical frequencies given... there's 3dB suppression at the critical frequencies... I assume 40dB/decade after that which means I need a 2nd order active filter with butterworth response?

Provide a diagram of the FDM transmitter. As a minimum, this need to include blocks for the LO, modulator (what type should you use), bandpass filter and amplifiers. Think about what you can use to combine different signals for before transmission.

Stuck here, I think my LO needs to be tuned to the channel frequencies I noted above. I am not certain about what i have said about the bandpass filter - but reasonably so.

-If the bandpass is required to supress the LSB as I have assumed than I can use a run of the mill DSBSC to modulate all the carriers and then just filter out the LSB before amplification??

Thanks for any help.

2. Aug 22, 2013

### rude man

Off to a rough start. I hope you haven't been reading my posts so far; I've deleted them all.

Thing is, if it's a ssb/sc system then there is no need to filter that signal. You can set the 1st carrier at 105 KHz, the 2nd at 110 KHz, the 3rd at 115 KHz and the 4th at 120 KHz. Since the max modulation frequency is only 4.5 KHz there would be no leakage into the next frequency "bin" even with no filtering.

This assumes the sideband occurring below the respective carrier freq.

Thing is, there are several ways of producing a ssb/sc output. E.g. by using phase shifters and balanced modulators, a filter is not required at all.

All of this makes me think they want you to come up with an actual ssb/sc modulator configuration which could consist of a dsb/sc modulator (a simple multiplier or balanced modulator) with the upperr sideband attenuated by at least 40 dB. Does that sound reasonable?

Last edited: Aug 23, 2013
3. Aug 23, 2013

### Fisher92

Thanks RM, haven't read any other posts - didn't even get an email for that one for some reason.

Firstly, Yes that does sound reasonable.

-I agree this is valid & gives better channel separation. In this case I will be keeping the LSB and discarding all the USB with my filters?

Since FDM in this case is effectively four individual systems with four distinct carriers and modulators (as I understand it anyway) to save time I will just go through what I am thinking for channel 1 (other channels are identical except shifted 5kHz).

Ring modulator is a good choice, (have a reasonable understanding of its operation) - the output of this 'block' is DSB-SC with LSB at 102.8kHz-105kHz and the USB at 105kHz to 107.2kHz. To get SSB-SC I want to use a band pass filter to block the upper side band and pass the LSB.

*The math from the OP applies-assuming it's correct? I get fc1=102.8kHz and fc2=105kHz ....f0=103.894kHz
Q=47.22...........Since this isn't really design per se, I guess they just want a block with 2nd order BP filter,Q and the critical frequencys - you can't really answer that last part, but if it makes sense to you?

The part of this question I have no idea about is 'Think about what you can use to combine different signals for before transmission.'..... From what I have read FDM (for radio at least) is essentially single systemps operating in the same band allotment and transmitted at the same time,,,, Is the reason I need to combine them just because they will be sent using the same antenna?

- If i were playing with actual electronics here (bear in mind I have not played with radios - this is coming from the experience I got building a polyphonic MIDI player back in the day) I would simply 'or' the signals together.. As I say that I realize it would only work with square waves - How can I combine the signals then? It can't be a case of simply joining the outputs of the amplifiers - wouldn't work.

In other news, I finally got feedback on the last assignment you helped me with..... 98% - not bad at all! The only problem with the questions I discussed with you was the problem you pointed out in your last post about the superhet question... everything else was spot on - THANKS AGAIN!

4. Aug 23, 2013

### rude man

Yes.
The sidebands extend from 0.1 KHz to 4.5 KHz, not 2.2 KHz.
The filters are low-pass, with a very sharp rolloff occurring at 100 Hz below the carrier frequency so that attenuation is 40 dB at 100 Hz above the carrier frequency. That is a vey tall order.

In fact, I claim that such a filter can't be a 2nd-order filter. I suspect you would need a high-order Chebyshev or similar type. Such a filter design is I believe addressed in a numer of Internet-available sites.

As I said, there is a much better way to synthesize a ssb/sc signal using dsb/sc modulators and phase shifters, which does not use filters to kill one sideband. But judging from the question you are to assume a filter.
Don't have any brilliant ideas right now. Certainly you can't just wire-or the four signals. But you can sum them in an op amp stage and then amplify the resulting signal & feed it to the antenna.

Good! I don't see the bit about 'superhet' operation. What was it?

5. Aug 23, 2013

### Fisher92

-The question was effectively explain the operation of the superhet but in parts.. One part was the BP filter BW necessary to eliminate the image problem. We ended up having 2fIF... Lecturers response to this was 'filter is centered around fc so it should be twice this....4fIF' Mistake cost me a whole mark (out of 100)! - Explanation of the image problem was apt so I wasn't really even penalized for the mistake.

..Back to the current problem, hopefully adjust my answer and repost soon

6. Aug 23, 2013

### rude man

Well, I see his point. In an actual receiver the filter would be centered at fc. That's because the filter gain should be max. at the carrier frequency for best s/n ratio. And a real filter rolls off to either side of its center frequency. Good point.

Then, say you're at fc = 1000 KHz, and the LO is at 1000 + 455 = 1455 KHz. So the input filter has to attenuate 1455 + 455 = 1910 KHz while peaking at 1000 KHz. And since it's symmetrical (almost) about the maxgain frequency = fc, the region below 1000 KHz has to be 910 KHz also, even though no signal in that range would make it through the IF stage. So 910 + 910 = 4fIF.

I assumed an ideal filter, unity gain end-to-end, in which case its width would only have to extend from 1000 KHz to 1910 KHz = 2 fIF.

7. Aug 23, 2013

### Fisher92

Yes, all the literature agrees - only reason I can see to force us to this methodology is that the other methods are conceptually more difficult. I suspect this is a question to prove that we understand what SSB-SC is. &FDM

Low Pass Filter on all four channels starting at the carrier frequency 105,110,115,120kHz. These frequency's are both my local oscillator frequencies and critical frequency for my low pass filters?

Why is the critical frequency at fc-100Hz... you say that the 40dB attenuation would then occur at fc+100Hz? But, the roll-off is dB/decade so at f(critical) +200Hz there is no guarantee that it's already at 40dB is there - I haven't done the maths behind this, curious as to how you came up with the fc-100 critical frequency for the filters?

My mistake, the 2.2kHz is from (4.5kHz-100Kz)/2 - I was thinking that the sidebands are made from 1/2 of the info BW on the USB & half on the LSB??? Back to my textbook.

I want a 40dB roll off after this critical frequency (by definition, the critical frequencies are at 3dB attenuation).

My textbook begs to differ-A second order filter (Butterworth response) by definition has a -40dB/Decade roll-off.. These theoretical definitions are not realized in application but should go close.

Since I can justify the roll-off for this, I was thinking of just using a run-of-the-mill sallen-key filter since there easy to design and according to my textbook meet the 40dB attenuation criteria??

Thoughts?

-Good explanation! & obviously the understanding I have now re superhets (thanks in no small part to you) is correct.

8. Aug 23, 2013

### rude man

I would like to forget about 'critical frequency' for a moment.

Let's look at what the filter has to do. Your dsb/sc spectrum extends from LSB = (fc-4500 Hz to fc-100Hz), then USB = (fc+100Hz to fc+4500 Hz). Your filter has to pass the LSB and reject the USB. That means it has to have essentially flat gain until fc - 100 Hz and drop off starting at fc - 100 Hz, reaching 40dB attenuation at fc + 100 Hz. That is a lot of attenuation in a very small number of decades!

A 40 dB/decade filter is useless. The no. of decades from fc - 100 Hz to fc + 100 Hz is, for fc = 120 KHz, = log10(120+0.1)/(120-0.1) = 7.24e-4 decade. So attenuation would be only 40dB*7.24e-4 = 0.029 dB and you need 40 dB! See what I mean about a radical filter design needed?
As I already explained, the highest part of your spectrum you need to include is at fc-100 Hz. By the time you get to fc+100 Hz you already have to be down 40 dB from the gain at fc-100 Hz. This is part of the specification in your problem statement.
No. You need a whole lot more rapid attenuation than 40 dB/decade. You need 40 dB/0.029 = 1379 dB/decade, starting at fc-100 Hz if you want to get to 40dB attenuation at fc+100 Hz.

This is why ssb/sc systems are not designed that way!

9. Aug 23, 2013

### Fisher92

Hmmmmmm... I definitely see your point here. - Good explanation.

I have an inkling that this is beyond the scope of the question - although definitely interesting so I will no doubt look into it regardless. - I will consult my lecturer about what it is he wants... The question explicitly states the 40dB roll off so it may be that I am required to look into this further/// Saturday here so there is no chance of talking to him.. Job for Monday morning.

-I can do Q and fc easily... fc(critical) is simply fc-100Hz for every channel and Q is whatever it turns out to be.

-There is also talk of amplifiers in the question. surely (short of actually designing the FDM transmitter) there is not enough info to give any specs on the amplifier?

Would this be a simple linear summer? Say 10K resistors at the output of each ring mod and into a single opamp? The resistors (again no real circuit info so its hard to give resistor values short of pulling them out of the air) are just to stop the outputs pulling the other outputs low/high?

Then I could just feed the output of the summer straight into another op-amp (or amplifier IC)? How would I go about choosing the gain here? ACMA (FCC equiv) must have limitations on transmitting power/ obviously this can't be for commercial use - op-amps don't come that big do they?

10. Aug 24, 2013

### rude man

Sticking for the moment with filtering (summing and amplifying come later, as do harware implementation), just to spoil your weekend there is a way to alleviate the demands on the lowpass filter.

Consider this: heterodyne the modulating signal with a low-frequency carrier first. This alleviates the demands on the LPF to filter out the USB. Then, take the output of that filter, dsb/sc it again with a suitable carrier frequency and do another LPF on the second USB. The output of the second LPF is your ssb/sc signal.

There exists a low heterodyning frequency such that the requirements on both filters is the same, namely the number of decades (still a small fraction) available to cause 40 dB attenuation. It involves the high end of the modulation spectrum, not the low.

11. Aug 24, 2013

### Fisher92

How would I determine the heterodyning frequency such that the requirements on the filters are them same.
Firstly I need to determine the lower order carrier (this is another oscillator, so I would end up with 8 LO, 4 analog mixers & 4 ring modulators in my design to get the 4 SSB-SC signals?)

*Does this also mean that my 105,110,115,120 carriers will need to be shifted? since the effective info signal into the ring mods is now a higher BW?

Thanks, - really confused, going to hit the internet.

12. Aug 24, 2013

### rude man

What to you is the difference between an analog mixer and a ring modulator? To me, both produce a dsb/sc signal.

Assuming that, the answer is that you would need just one low-frequency LO, 4 high-freq LO's, 8 mixers (multipliers, ring modulators, whatever - dsb/sc generators) and 8 low-pass filters.

The way I set it up your virtual carriers would be at 100, 105, 110 and 115 KHz. But there are various combinations possible.

Did you understand how I derived the rolloff requirements for the one-filter scheme? Assuming yes, the next step is for you to assume a low-frequency LO. I will give you a suggestion: 6 KHz. Filter the USB from that modulator so that the input to the 2nd modulator sees the LSB only. What is the dB rolloff requirement of the 1st LP filter now? Compare with the one-filter scheme. Then try to pick a frequency somewhere near the four virtual carriers, i.e. around 100 KHz, so as to give you the spectrum you need for it to be a real SSB/SC output. What is the dB rolloff requirement of the 2nd LP filter?

The result I got was that the LSB of the 2nd modulator output is in effect the USB of 100Kz.

13. Aug 25, 2013

### Fisher92

To me a mixer is not necessarily DSB-SC - Could be DSB-FC or anything, generic term for a modulator - I was thinking that I wanted DSB-FC for this part of the question - From your answer it is clear that is not the case though.

One LO @ 6kHz to be used as a '1st stage' carrier will give all four information signals as (assuming suppressed carrier) - This gives me information signals effectively of 1.5kHz-5.9kHz & 6.1kHz-10.5kHz
Good Question.
*Assuming I need a 40dB minimum rolloff at 6.1kHz then I need
$$40dB=rolloff*log(\frac{6.1}{5.9}=2762.85$$ This is still a radical filter... I could get maybe 200dB/decade from simple analog filters but this is just not going to happen?!

14. Aug 25, 2013

### rude man

As far as I know you never go to dsb/fc unless you want to broadcast this type of signal. It's done because demodulation is easy via an envelope detector. Which is why the 540 - 1600 KHz bdcst band is that way.
It might be doable but you'd need many 40dB/decade stages obviously. That's why ssb/sc is not done that way!

Anyway, you can do a bit better with the 1st filter and make it relatively easy to come up wit the second if you do it this way instead (for the 1st band):

1st dsb/sc modulation with 4.5 KHz; wipe out the LSB with a high-pass filter set to roll off at and below 4.6 KHz, then feeding the USB into a 2nd dsb/sc modulator set to 95.5 KHz. The 2nd mod output is then also hi-pass filtered to eliminate its LSB.

Your new basebands are at 100, 105, 110 and 115 KHz, with the LSB's suppressed.

Why don't you analyze this. The 1st filter is still a bear but a bit better than before, while the 2nd is much easier to implement.

I see no way to avoid coming up with a hi-pass filter set to roll off at 4.6 KHz, reaching 40 dB attenuation at 4.4 KHz, unfortunately. As I said, it's not done that way except in classrooms!

So instead of 8 lo-pass filters you'd go with 8 hi-pass ones. Hi-pass fiters are no harder (or easier) to come by than lo-pass.

15. Aug 25, 2013

### rude man

I found a pretty good paper on Chebyshev filter design (appended herewith).

As you can read, there is a tradeoff between the "ripple" (gain variation) within the passband and the required order of the filter.

Looks to me like a 4.5 KHz LSB-suppression hi-pass Chebyshev filter could work with about order 10 (that would be ~5 op amp stages I believe).

The paper does not give hardware design procedure; software like Nuhertz can do that (but unfortunately it's not free for this job).

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16. Aug 25, 2013

### Fisher92

Hey RM, my lecturer got back to me on this-

1. You do not need to actually design the filter.
2. If you wanted to design a filter with this high a rolloff you would need crystal or ceramic filters.
3. Just work out Q & fc and use them as blocks in the system diagram.
4. Low pass filter is Okay, but could be trouble if there are some other frequency components in the information signal. - Safer to go with BP filter and select exactly the frequency range you want.
5. For mixing the channels, any linear mixer will do -- op amp is a solution .... Again, you can just add a block called 'linear summer' and leave it at that.

Makes the Question substantially easier. I'm going to try and get a block diagram done now.... I'f you wouldn't mind checking what I come up with>

Thanks

17. Aug 25, 2013

### rude man

I am very anxious to see your block diagram!

18. Aug 25, 2013

### Fisher92

Missed something very obvious in my COMMS textbook when we started to design the actual filter... Attached because I thought you might find it interesting - there is also a textbook example of SSB-SC with a filter to eliminate a sideband.... Swear I went through this when I first started the question and then completely forgot about it...

LO's:
Channel 1: 105kHz
Channel 2: 110kHz
Channel 3: 115kHz
Channel 4: 120kHz

These are then fed with the information signal into 4 ring modulators....
The output of the ring modulators is DSB-SC with frequency components at:
Channel 1: LSB(100.5-104.9) USB(105.1-109.5) kHz
Channel 2: LSB(105.5-109.9) USB(110.1-114.5) kHz
Channel 3: LSB(110.5-114.9) USB(115.1-119.5) kHz
Channel 4: LSB(115.5-119.9) USB(120.1-124.5) kHz

Now using either ceremic or crystal filters I need to remove the USB.

According to my textbook example, fc is the carrier frequency (not critical), LO, and I end up with:
$$Q=\frac{f_c(log^{-1}dB/20)^{1/2}}{4Δf}$$
Channel 1:
$$Q=\frac{105k(log^{-1}*40/20)^{1/2}}{4*200}=1312.5$$
Channel 2:
$$Q=\frac{110k(log^{-1}*40/20)^{1/2}}{4*200}=1375$$
Channel 3:
$$Q=\frac{115k(log^{-1}*40/20)^{1/2}}{4*200}=1437.5$$
Channel 4:
$$Q=\frac{120k(log^{-1}*40/20)^{1/2}}{4*200}=1500$$

The block will be Filter, Q,fc four times..

The only problem here is that I have left out any amplifiers? Should I amplify the signal, the out put?? or somewhere in the middle?

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19. Aug 26, 2013

### rude man

OK, I'm not familiar enough with the crystal or ceramic filters your textbook discusses. So that's where the Q values are derived from!

I do wonder how flat the bandpass filters are. In the example, a 9MHz +/- 2 KHz dsb/sc signal has one of its sidebands filtered out. I hope the bandpass filter is reasonably flat over its passband, otherwise gross distortions of signals in the (in your case) 0.1 - 4.5 KHz range would obtain.

Interesting that the textbook scheme also uses double modulation, except they use 9 MHz as the 1st LO to accommodate the ceramic or crystal filter, then the 2nd IF gets you the desired antenna spectrum.

I guess the case is closed, but I do wonder about the flatness of those ceramic or crystal b/p filters. I would have expected a peak around 2.25 KHz (center) away from 9 MHz and a sharp rolloff going to 0.1 KHz and 4.5 KHz so the required attenuation only 200 Hz away is realized. Which as I say would cause monstrous distortions.
If you can get any info on the filters intended for this appilcation I would love to see the frequency responses. Maybe ask your lecturer?

20. Aug 26, 2013

### Fisher92

I thought that was interesting too. I don't know if there is any reason for me to use double modulation for this question as it is not really a design question?

Whould you amplify the information signal as the textbook does? No information on the nature of the signal makes it hard but there must be a standard design that place the amplifiers somewhere?

Thanks