#### What is in this article?:

- Spectrum Inversion Ensures Compliance With 3GPP2
- A Plan Of Action

Spectrum inversion is used in 3GPP2 communications systems and can be implemented with various transceivers through the use of three simple methods.

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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)