In ±3QMAS NMR of a spin I = 5/2 system, an echo signal E(t_{1}, t_{2}) and an antiecho signal
A(t_{1}, t_{2}) have the following expressions:

- E(t
_{1}, t_{2}) = exp(-i ω_{1}t_{1}) exp(i ω_{2}t_{2}) - A(t
_{1}, t_{2}) = exp(+i ω_{1}t_{1}) exp(i ω_{2}t_{2})

_{1}is the evolution period and t

_{2}the acquisition period.

## Echo and antiecho in z-filter MQMAS

Fig. 1: Z-filter 3QMAS NMR pulse sequence and coherence transfer pathway.

The echo amplitude and the antiecho amplitude have the
same sign.

The three-pulse z-filter MQMAS sequence generates echo and antiecho, which have the
**same amplitude** with the **same sign**.

The detected signal is the contribution of E(t_{1}, t_{2}) and
A(t_{1}, t_{2}):

S_{3}(t_{1}, t_{2}) = H.E(t_{1}, t_{2})
+ H.A(t_{1}, t_{2})

The amplitude H is a complex number. The 2D Fourier transform of
S_{3}(t_{1}, t_{2}) generates phase-twisted 2D peak.

The standard solution to this problem is the application of States procedure to generate 2D pure absorption peak. It consists in acquiring two complementary signals:

- S
_{x}(t_{1}, t_{2}) = S_{3}(t_{1}, t_{2}) = H.cos(ω_{1}t_{1}) exp(i ω_{2}t_{2}) - Sy(t
_{1}, t_{2}) = H.cos(ω_{1}t_{1}**+ π/2**) exp(i ω_{2}t_{2}) = -H.sin(ω_{1}t_{1}) exp(i ω_{2}t_{2})

The echo S_{E}(t_{1}, t_{2})
and antiecho S_{A}(t_{1}, t_{2}) signals
are obtained from S_{x}(t_{1}, t_{2})
and Sy(t_{1}, t_{2}):

- S
_{E}(t_{1}, t_{2}) = Sx(t_{1}, t_{2}) + i.Sy(t_{1}, t_{2}) - S
_{A}(t_{1}, t_{2}) = Sx(t_{1}, t_{2}) - i.Sy(t_{1}, t_{2})

## Echo and antiecho in two-pulse MQMAS

Fig. 1: Amplitude-modulated two-pulse 3QMAS NMR pulse sequence and coherence
transfer pathway.

The echo amplitude and the antiecho amplitude have
opposite signs.

The two-pulse MQMAS sequence generates echo and antiecho, which do not have the same amplitude but with opposite signs.

The detected signal is the contribution of E(t_{1}, t_{2}) and
A(t_{1}, t_{2}):

S_{2}(t_{1}, t_{2}) = H.E(t_{1}, t_{2})
- H.A(t_{1}, t_{2})

For simplicity, they are supposed to have **opposite amplitudes**.
The 2D Fourier transform of S_{2}(t_{1}, t_{2})
generates phase-twisted 2D peak.

The States procedure consists in acquiring two complementary signals:

- S
_{x}(t_{1}, t_{2}) = S_{2}(t_{1}, t_{2}) = - i.H.sin(ω_{1}t_{1}) exp(i ω_{2}t_{2}) - Sy(t
_{1}, t_{2}) = - i.H.sin(ω_{1}t_{1}**+ π/2**) exp(i ω_{2}t_{2}) = -i.H.cos(ω_{1}t_{1}) exp(i ω_{2}t_{2})

The echo S_{E}(t_{1}, t_{2})
and antiecho S_{A}(t_{1}, t_{2}) signals
are obtained from S_{x}(t_{1}, t_{2})
and Sy(t_{1}, t_{2}):

- S
_{E}(t_{1}, t_{2}) = Sx(t_{1}, t_{2}) + i.Sy(t_{1}, t_{2}) - S
_{A}(t_{1}, t_{2}) = Sx(t_{1}, t_{2}) - i.Sy(t_{1}, t_{2})

## Conclusion

For these two pulse sequences in ±3QMAS NMR of a spin I = 5/2 system,
the echo S_{E}(t_{1}, t_{2})
and antiecho S_{A}(t_{1}, t_{2}) signals are obtained
with the same formulas:

- S
_{E}(t_{1}, t_{2}) = Sx(t_{1}, t_{2}) + i.Sy(t_{1}, t_{2}) - S
_{A}(t_{1}, t_{2}) = Sx(t_{1}, t_{2}) - i.Sy(t_{1}, t_{2})

These two formulas are a consequence of the +π/2 angles involved in
Sy(t_{1}, t_{2}). This +π/2 angle is included in the
pulse program by applying to the first pulse a phase change of +π/|2p|
for Sy(t_{1}, t_{2}) acquisition.

Conversely, Sx(t_{1}, t_{2}) and Sy(t_{1}, t_{2})
can be expressed as:

- Sx(t
_{1}, t_{2}) = [S_{E}(t_{1}, t_{2}) + S_{A}(t_{1}, t_{2})]/2 - i.Sy(t
_{1}, t_{2}) = [S_{E}(t_{1}, t_{2}) - S_{A}(t_{1}, t_{2})]/2

In the States method, pure 2D absorption spectrum is the Fourier transform in
the t_{1} domain of

Re[Sx(t_{1}, ω_{2})] + i.Re[Sy(t_{1}, ω_{2})]

(top)