For example, if you have a QAM 16, which mean there are 16 symbols.  Each symbol will have different level of the QAM constellation.  Each symbol will have a corresponding match filter.  At the receiver, there is a bank of match filter to match with each received symbol.

The receiver will use threshold to detect which has the highest threshold of that matched filter array.  My question is, let's say you have two symbols: one with high level and one with lower level.  Is it guaranteed that only the symbol that got "matched" will have the highest threshold level after the matched filter output?

I mean, can the high level symbol have a higher output after the matched filter even if it is not matched vs. a lower level symbol that is matched?

Westrace, perhaps I am guilty of requiring you to be too precise, but usually the matched filter is used to identify a signal buried in noise. Unless the noise is guaranteed to be sufficiently weaker than the signal, there is no possible guarantee that a matched filter for signal 1 will have a higher output, when signal 1 is transmitted, than the matched filter for, say, signal 2. It has to be a probabilistic comparison.

My question was more in theory, even in the absent of noise added.  

For example if I have two symbols: 

symbol one has an amplitude of A so its matched filter is : A s1(Ts-t)

symbol two has an amplitude of A/4 so its matched filter is: A/4 s2(Ts-t)

Now if symbol two is transmitted, is it possible if the output of matched filter 1 can have a higher output because the matched filter 1 has a higher amplitude vs. matched filter number 2?  What we want is the output of matched filter 2 to have a higher amplitude if symbol two is transmitted.  I mean when you convolved the matched filter to the received symbol, if the matched filter has an inherent high amplitude then its output will be higher regardless of the received symbol.

In a QAM16 or QAM32 …,  where each symbol will have different amplitude, is it true that, when a given symbol is transmitted, only the matched filter for that symbol will have the highest output?

In any modulation system for which the constellation of possible signals, if there are several constellation members with the same angle but different amplitudes you have to be concerned that the channel will change the amplitude overall and one constellation member transmitted will look like another constellation member when received. Similarly constellation members can be confused with one another if the channel has other kinds of distortions like phase shift, frequency shift, multipath, etc. 

In practice, such systems use an adaptive method to adjust compensations for the possible channel distortions (OTHER THAN NOISE) so that after compensation the received signal closely matches the transmitted signal. The demodulation then chooses the received demodulated signal to be the one closest to a constellation point. If the compensation is pretty good, the closest constellation point will be correct. If the compensation is perfect, the demodulated signal will perfectly coincide with one constellation point. 

In practice, there is always some noise and it has two separate effects. The obvious effect is that the noise makes the received demodulated signal not coincide with a constellation point and for a large enough instance of the noise, the demod signal can be closer to another constellation point. The other effect is that the noise can corrupt the compensation so that constellation points are displaced - then even a momentarily noiseless received signal could be closer to a wrong constellation point.

With QAM type signals there is another issue. Several transmitted signals are sent at the same time, but time shifted, e.g. one "chip" can be sent in 0 to T   and another can be sent in T/2 to 3T/2.  Then the received constellation is not a set of say 16 points but rather is an "eye diagram". 

Every symbol has same shape in principle no matter what level it has in IQ map. If a symbol in 16QAM is lower than another by design then you want to keep it like that. 

I am familiar with one filter for the stream of symbols.

I am not sure what is that bank of correlators for. But my guess is that if you view each sequence of symbols as a unique shape then we might have different correlators e.g for 16QAM we have 4 levels on I or Q and that means a variety of sequences. But such approach seems unrealistic since one filter is well known to work.  

Usually there's only one matched filter in the receiver, that is matched to the filter shape in the transmitter.   Signal levels coming out of that matched filter in the receiver are "sliced" to see which level of the constellation they are closest to.

The easiest example is a BPSK or QPSK receiver, where the coordinate axes are halfway between the constellation points, so if the received signal level is above or below the axis the hard decision level is either a 1 or 0.

Likewise for multi-level constellations like 16-QAM, a decision line is drawn halfway between the constellation points that determines which point the signal will be decoded as.

So, only one matched filter, and take the constellation point closest to the received signal level as the most likely point transmitted.

Thank you for your reply.  There are still a lot of literatures that talk about a bank of matched filter.  For example, on Proakis, Digital Communication 5th Edition, Fig. 4.2 - 7 and Fig. 4.2 - 8, depict a bank of matched filter for each symbol.  It seems to imply there is one matched filter per transmitted symbol.  Please see the inserted file for the image.  

I also notice that some other literatures have only one matched filter, one for the inphase and one for the quadrature.  So I am a bit confused as to which one is being used.  

Here is one example in which only one matched filter is used.

I think you are misunderstanding the figure in Proakis.   I have the 4th Ed which doesn't have that figure, so can't comment directly on what might be being shown there, but QAM receivers are not built this way.   Proakis shows a a lot of diagrams that are meant for theory, not for implementation, and I suspect that's either what this is, or something else is being shown.   I see that the correlator correlates against M symbols, so it might be a CDMA or spread spectrum receiver or something, I really don't know.

If the channel has no memory then we can do symbol by symbol demodulation.One matched filter which will match to transmitted pulse shape(which is common for all 16 QAM points) will be used.This filter maximises SNR.The filter output will be given to nearest neighbour detecor where 16 points of the QAM constellation contend.Winner will be the one which is least Euclidien distance to matched filter output.

If the channel has memory then IntersSymbol Interference (ISI) will be there.We can not detect symbol by symbol.We have to consider the entire sequence and use Vitebi or ML detector.