In case of QAM and other passband modulation schemes, Rx has no information about carrier phase of the Tx oscillator. To see the effect of the carrier phase offset, consider that a transmitted passband signal consists of two PAM waveforms in and arms denoted by and respectively and combined as

(1)

Here, is the carrier frequency and and are the continuous versions of sampled signals and given by

(2)

In the above equation, and are the inphase and quadrature components of the symbol, are the samples of a square-root Nyquist pulse with support and and are the sample time and symbol time, respectively.

In the absence of noise, the received signal for a passband waveform is the same as the transmitted signal except a carrier phase mismatch, i.e., we ignore every other distortion in the received signal except the phase offset .

After bandlimiting the incoming signal through a bandpass filter, it is sampled by the ADC operating at samples/second to produce

where the relation is used and corresponds to , and the angle is the phase difference between the incoming carrier and the local oscillator at the Rx.

To produce a complex baseband signal from the received signal, the samples of this waveform are input to a mixer which multiplies them with discrete-time quadrature sinusoids in the arm and for arm. We continue the derivation for part and the same for is very similar and the reader can solve it as an exercise. Using the identities and ,

The matched filter output is written as

The double frequency terms in the above equation are filtered out by the matched filter , which also acts a lowpass filter due to its spectrum limitation in the range , where is the symbol rate. Writing the definitions of and from Eq (2),

where comes into play from the definition of auto-correlation function. To generate symbol decisions, -spaced samples of the matched filter output are required at . Downsampling the matched filter output generates

For a square-root Nyquist that satisfies no-ISI criterion, is zero except for . Thus,

A similar derivation for arm yields the final expression for the symbol-spaced samples in the presence of phase offset .

From the phase rotation rule of complex numbers, we know that this expression is nothing but counterclockwise rotation by an angle . In polar form, this expression can be written as

In conclusion, a mismatch of between incoming carrier and Rx oscillator rotates the desired outputs and on the constellation plane by an angle . This is drawn in the scatter plot of Figure below for a -QAM constellation. Keep in mind that the blue circles are not one but several symbols mapped over one another due to the similar phase shift.

In Eq (3), start with and observe that the and outputs are and , respectively. This implies that signals in and arms are completely independent of each other. Gradually increasing has two effects:

- Since , amplitude of in reduces. The same phenomenon happens with in .
- Since , interference of in increases as well as that of in .

This interference between and components is known as cross-talk. Cross-talk increases with until for a difference, appears at output and at output.

The effect of this cross-talk on a Raised Cosine shaped -QAM waveform with excess bandwidth is shown in Figure below for a phase difference . Observe the first sample: it is in quadrant II. After phase rotation, part moved towards left thus increasing its amplitude and moved downwards reducing its amplitude. This is evident through the first samples in the Figure. A similar argument holds for all other symbols.

Something really interesting has happened in Figure above. Notice that although the amplitude has decreased for some symbols, it has risen for some other symbols as well. This is the outcome of a circular rotation. While it is good to have some symbols with a little extra protection against noise, remember that it has come at a cost of reduced amplitudes for other symbols, making them much vulnerable to noise and other impairments. The overall effect is negative, just like strengthening your right arm in exchange of significantly weakening the left is dangerous for your body.

What was discussed above can be extended to the whole symbol stream. The cumulative effect of a phase offset is straightforward to see in a scatter plot. There will be clouds of samples from downsampled matched filter output around the original constellation.

Since the scatter plot is different than a raw time domain waveform, we employ the eye diagram to examine the effects of carrier phase offset (say, on an oscilloscope). First, start with a BPSK modulation scheme and remember that there is no channel in this case and consequently no cross-talk. However, the effect of phase rotation is a reduction in signal amplitude which can be observed by plugging in Eq (3) and only focusing on arm.

(5)

Since always lies between and , the amplitude of the signal gets reduced accordingly with the rest of the energy rising in the arm. From Eq (3), this signal is written as

With a phase offset of , the branch loses half of its energy with the remaining half going in the arm. This is drawn in Figure below. In fact for a phase rotation, the contribution actually reaches zero and all the energy of the signal appears across the branch. Due to this reason, we will see later that the arm is still employed for BPSK signals — not for data detection but helping in the phase synchronization procedure.

Next, we turn our focus towards QAM and observe the amplitude change and cross-talk between and branches for three different phase offsets, , and . Figure~?? illustrates the channel for these phase offsets in a noiseless case and a -QAM signal. A similar diagram holds for arm as well and not drawn here. The optimal sampling instants are still visible due to zero noise but \bbf{the eye diagram looks more like a -PAM signal than that of a single -QAM signal due to the cross-talk from arm}. It is also evident that and affect each other in equal proportions.

The reason there are only two eyes for phase offset is that and in Eq (3) become equal and hence many symbols and cancel each in both and arms of the output. In terms of the scatter plot, a rotation of shifts the constellation points onto the real and imaginary axes, so for the plot shown here, the output at the sampling instant coincides only with a positive or negative symbol value.