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Spectrum inversion is part of the physical layer in Third Generation Partnership Program 2 (3GPP2) using NCDMA techniques, prior to transmissions and following signal reception. Due to the large number of available RF transceivers and baseband processors on the market, it is easy to see how RF transceivers and baseband processors could easily have mismatched spectrum in both the transmit and receive paths. Such a simple oversight will result in noncompliance with the 3GPP2 standard and failure to achieve signal demodulation. Fortunately, for those involved with 3GPP2 systems, a few simple techniques can help determine if spectrum inversion has been performed on a signal. Three simple techniques are also presented to help perform spectrum inversion on RF transceivers that do not have built-in spectrum inversion.

In a transmitter, the easiest way to determine whether the spectrum has been inverted is by comparing a single-tone continuous-wave (CW) wave with nominally positive frequency to the local-oscillator (LO) frequency. If the RF CW output frequency is greater than the LO frequency (a positive offset), then no spectrum inversion has occurred. But if the LO frequency is greater than the RF output frequency, the spectrum has been inverted.

In the receive path, if a positive offset RF input frequency produces an in-phase (I) output that leads the quadrature (Q) output by 90 deg., then no spectrum inversion was performed by the RF demodulator. Generally, the modulation format of the RF demodulator follows the modulator. These points can be demonstrated by examining the uplink and downlink paths of a WCDMA system, as specified in the 3GPP standard TS 25.213 **(Fig. 1)**. For simplicity’s sake, assume that the transmitter baseband I and Q signals are represented by the expression:

V_{m} = e^{jω}_{m}t = cos(ω_{m}t) + jsin(ω_{m}t)

where this is a positive frequency, yielding a complex tone at baseband. The transmitter’s I and Q LO signal components are represented by the expression:

LO_{ITX} = cos(ω_{m}t) and

LO_{QTX} = -sin(ω_{m}t)

Next, notice the negative polarity of the Q LO signal:

V_{TX} = cos(ω_{m}t)cos(ω_{LO}t) - sin(ω_{m}t)sin(ω_{LO}t)

V_{TX} = 0.5cos[(ω_{m} - ω_{LO})t] + 0.5cos[(ω_{m} + ω_{LO})t] - 0.5cos[(ω_{m} - ω_{LO})t] +(ω_{m} + ω_{LO})t]

and

V_{TX} = cos[(ω_{m} + ω_{LO})t]

As these expressions reveal, a positive modulation baseband signal, in combination with a negative-phase LO frequency, produces an RF output frequency which resides above the LO frequency. The result is no spectrum inversion.

On the receiving end, assume that the same transmitted RF signal is received and demodulated with the same LO format as the transmitter:

V_{RX} = cos(ω_{RX})t

with

ω_{RX} = ω_{m} + ω_{LO}

V_{LO} = e^{-jω}_{LO}t = cos(ω_{LO}t) - jsin(ω_{LO}t)

V_{BB} = cos(ω_{RX}t)cos(ω_{LO}t) - jcos(ω_{RX}t)sin(ω_{LO}t)

After multiplying out the terms and applying lowpass filtering to remove higher-frequency signal components:

V_{BBI} = 0.5cos[(ω_{RX} - ω_{LO})]t]

and

V_{BBQ} = 0.5sin[(ω_{RX} - ω_{LO})]t]

Substituting ω_{RX} with ω_{m} + ω_{LO}, the I and Q baseband outputs are the same as the I and Q transmit baseband inputs:

I_{m} = cos(ω_{m}t)

and

Q_{m} = sin(ω_{m}t)

## A Plan Of Action

With a clearer understanding of how to detect spectrum inversion, it is now time to explore available methods for achieving it. As mentioned earlier, the 3GPP2 standard requires that the signal spectrum of NCDMA be inverted prior to transmission and after reception. Three simple methods can be used for spectrum inversion.

In the first method, spectrum inversion is achieved by adopting a positive phase LO. This is the modulation format recommended by the 3GPP2 standard **(Fig. 2)**:

I_{m} = cos(ω_{m}t)

and

Q_{m} = sin(ω_{m})

and

LO_{ITX} = cos(ω_{LO}t)

and

LO_{QTX} = sin(ω_{LO}t)

V_{TX} = cos(ω_{m}t)cos(ω_{LO}t) + sin(ω_{m}t)sin(ω_{LO}t)

V_{TX} = 0.5cos[(ω_{m}) - ω_{LO})]t + 0.5cos[(ω_{m} + ω_{LO})]t + 0.5sin[(ω_{m} - ω_{LO})]t + 0.5sin[(ω_{m} + ω_{LO})]t

V_{TX} = cos[(ω_{m}) - cos(ω_{LO})t]

As shown here, the RF output frequency is lower than the LO frequency. Therefore, the spectrum has been inverted.

In method 2, spectrum inversion is achieved by inverting the polarity of the Q baseband signal:

I_{m} = cos(ω_{m}t)

and

Q_{m} = -sin(ω_{m}t)

LO_{ITX} = cos(ω_{LO}t)

and

LO_{QTX} = -sin(ω_{LO}t)

V_{TX} = cos(ω_{m}t)cos(ω_{LO}t) + sin(ω_{m}t)sin(ω_{LO}t)

V_{TX} = 0.5cos[(ω_{m}) - ω_{LO})]t + 0.5cos[(ω_{m} + ω_{LO})]t + 0.5cos[(ω_{m} - ω_{LO})]t - 0.5cos[(ω_{m} + ω_{LO})]t

V_{TX} = cos[(ω_{m} - ω_{LO})t]

In the third method, spectrum inversion is achieved by swapping the I and Q transmit baseband signals:

I_{m} = sin(ω_{m}t)

and

Q_{m} = cos(ω_{m}t)

LO_{ITX} = cos(ω_{LO}t)

and

LO_{QTX} = -sin(ω_{LO}t)

V_{TX} = cos(ω_{m}t)cos(ω_{LO}t) + sin(ω_{m}t)sin(ω_{LO}t)

V_{TX} = 0.5sin[(ω_{m}) + ω_{LO})]t + 0.5sin[(ω_{m} - ω_{LO})]t - 0.5sin[(ω_{m} + ω_{LO})]t + 0.5sin[(ω_{m} - ω_{LO})]t

V_{TX} = cos[(ω_{m} - ω_{LO})t]

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Method 1 is the approach recommended by the 3GPP2 standard for performing spectrum inversion. However, as shown above, other methods can achieve the same result and still be compliant with the specifications. In all cases for parts derived from the MAX2553 WCDMA transceivers from Maxim Integrated™, the baseband input and output pins are labeled so that no spectrum inversion occurs in the transceiver—either in the transmitter or receiver path. For compliance with the 3GPP2 standard, one should be aware of this and apply one of the above methods for inverting the transmit spectrum. For the receive path, a corresponding inversion is also required in most cases, unless the baseband is already programmed to accept inverted spectrum signals.