Patent Publication Number: US-2007099582-A1

Title: Method and apparatus for signal demodulation and transmission

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
      This application claims the benefit of U.S. Provisional Application No. 60/1731697, filed Oct. 31, 2005. 
    
    
     BACKGROUND OF THE INVENTION  
      1. Field of the Invention  
      The invention relates to dual-band transceivers, and in particular, to a synthesizer supporting dual frequency bands for a transceiver.  
      2. Description of the Related Art  
       FIG. 1  shows a conventional dual-band transceiver. As known, IEEE 802.11 a standard utilizes 5 GHz band whereas IEEE 802.11 b/g utilize 2.4 GHz band. The transceiver may be inside a WLAN device supporting multiple modes and standards. In  FIG. 1 , the 5 G demodulator  102   a , 2.4 G demodulator  102   b  and common IF demodulator  104  form a dual-band receiver. The 5 G demodulator  102   a  and 2.4 G demodulator  102   b  individually receive an RF signal RF a  of 5 GHz and an RF signal RF b  of 2.4 GHz, and demodulate them into intermediate signal IF a  and intermediate signal IF b  along two different RF paths IF a  and IF b  can be signals with a common intermediate frequency. A 5 G synthesizer  110   a  and a 2.4 G synthesizer  110   b  are required for the first demodulation step, each providing oscillation sources of the corresponding RF a  and RF b  frequencies and down-converting the RF signal into the intermediate signal IF a  and IF b  with the same frequency. The common IF synthesizer  120  generates signal for the IF demodulator  104  which demodulates the intermediate signal IF a  and IF b  to baseband, generating an inphase signal BB I , and a quadrature signal BB Q . The baseband signals, BB I , and BB Q , can be either in analog or digital format. The two stage demodulation described may use a super heterodyne architecture. Similarly, for signal transmission, the 5G modulator  112   a , 2.4G modulator  112   b  and IF modulator  114  form a dual-band transmitter which modulates the baseband signals and transmits the first RF signal RF a  and second RF signal RF b . The same frequency synthesizers as in the receiver can be re-used in the transmitter. Thus a total of three independent synthesizers may be required in a dual band transceiver as shown in  FIG. 1 .  
      In this architecture, optional external low-noise amplifiers (LNAs), variable gain amplifiers (VGAs) may have to be used to enhance the receiver sensitivity and power amplifiers (PAs) may have to be used to boost the transmitter output power. Various high-pass, low-pass and polyphase filters may be necessary for channel selection and image rejection. The costs of these components are considerable and known as a design issue. Additionally, implementing multiple synthesizers in a single chip is area-expensive and has potential signal interference problem. Therefore an improvement for dual-band modulation and demodulation is desirable.  
     BRIEF SUMMARY OF THE INVENTION  
      An exemplary embodiment of the transceiver comprises a transmitter and a receiver. The receiver comprises an RF demodulator and an IF demodulator. The RF demodulator is capable of receiving both the first RF signal of a first frequency and the second RF signal of a second frequency individually. The RF demodulator down-converts the first or second RF signal into an intermediate signal utilizing the first oscillation signal. Then the IF demodulator converts the intermediate signal into an inphase baseband signal and a quadrature baseband signal utilizing the second oscillation signal. A synthesizer is provided, generating the first and second oscillation signals from a single oscillation reference signal. Each of the first and second oscillation signals comprises an inphase part and a quadradure part. The first RF frequency is the sum of the frequency of the first oscillation signal and the frequency of the second oscillation signal, and the second RF frequency is the difference of the frequency of the first oscillation signal and the frequency of the second oscillation signal.  
      The frequency synthesizer may comprise an oscillator, and three dividers. The oscillator generates a reference frequency. The first divider is coupled to the oscillator, receiving the oscillation reference signal and dividing the reference frequency by a first value to generate the first oscillation signal with both the inphase and quadrature oscillation signals LO 1   I  and LO 1   Q . The inphase divider, coupled to the first divider, receives the inphase part of the first oscillation signal LO 1   I  and divides the frequency thereof by a second value to generate an inphase part of the second oscillation signal LO 2   I . The quadrature divider, coupled to the first divider, receives the quadrature part of the first oscillation signal LO 1   Q  and divides the frequency thereof by the second value to generate a quadrature part of the second oscillation signal LO 2   Q . Specifically, the first value is 2 and the second value is 3.  
      Embodiments of RF and IF demodulators are provided. Additionally, the transmitter in the transceiver comprises an IF modulator and an RF modulator. The IF modulator converts the inphase and quadrature baseband signals to the intermediate signal utilizing the second oscillation signal. The RF modulator, coupled to the IF modulator, up-converts the intermediate signal to the first RF signal or the second RF signal utilizing the first oscillation signal. The first or second RF signal is transmitted after frequency conversion. Various embodiments of the RF modulator and IF modulator are also provided. Furthermore, embodiments of the signal demodulation method, signal transmission method implemented by the transceiver are also provided.  
      A detailed description is given in the following embodiments with reference to the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      The invention can be more fully understood by reading the subsequent detailed description and examples with references made to the accompanying drawings, wherein:  
       FIG. 1  shows a conventional dual-band transceiver;  
       FIG. 2   a  shows an embodiment of a dual-band transceiver in accordance with the current invention;  
       FIG. 2   b  shows an embodiment of a frequency distribution for dual-band demodulation;  
       FIG. 3  shows an embodiment of synthesizer  202  in  FIG. 2   a;    
       FIG. 4   a  shows an embodiment of a RF demodulator  210  and a IF demodulator  212 ;  
       FIG. 4   b  shows an embodiment of a RF modulator  220  and a IF modulator  222 ;  
       FIG. 5   a  shows another embodiment of the RF modulator  220  and IF modulator  222 ; and  
       FIG. 5   b  shows a further embodiment of the RF modulator  220  and IF modulator  222 . 
    
    
     DETAILED DESCRIPTION OF THE INVENTION  
      The following description is of the best-contemplated mode of carrying out the invention. This description is made for the purpose of illustrating the general principles of the invention and should not be taken in a limiting sense. The scope of the invention is best determined by reference to the appended claims.  
       FIG. 2   a  shows an embodiment of a dual-band transceiver. An RF demodulator  210  receives a first RF signal RF a  and a second RF signal RF b . When either signal is received, the RF demodulator down-converts the received signal to an intermediate signal IF utilizing a first oscillation signal LO 1 . The same intermediate IF frequency can be obtained from both RF bands by selecting the frequency of LO 1  to be the arithmetic mean of RF a  and RF b . The IF demodulator  212  then converts the intermediate signal IF to an inphase baseband signal I(t) and a quadrature baseband signal Q(t) utilizing a second oscillation signal LO 2 . The frequency of LO 2  is half of the difference of RF a  and RF b . The inphase and quadrature baseband signals I(t) and Q(t) are then output in either analog domain for further data conversion or in digital domain for further digital signal processing. In this way, both the RF a  and RF b  signals are down converted to an identical IF and then baseband frequency using common hardware.  
      A transmitter in the dual-band transceiver comprises an RF modulator  220  and an IF modulator  222 , performing signal modulation similar but reversed in function to the demodulation process performed by the RF demodulator  210  and IF demodulator  212 . The IF modulator  222  up-converts the inphase baseband transmit signal I(t) and quadrature baseband transmit signal Q(t) to the intermediate signal IF utilizing second oscillation signal LO 2 . The RF modulator  220 , coupled to the IF modulator  222 , up-converts the intermediate signal IF to the first RF signal RF a  or the second RF signal RF b  utilizing the first oscillation signal LO 1 , and the converted first RF signal RF a  or second RF signal RF b  are then transmitted thereby.  
       FIG. 2   b  shows an embodiment of frequency distribution for dual-band demodulation. In order to down-convert RFa and RFb to the same intermediate frequency for the subsequent demodulation step, LO 1  and LO 2  are chosen to have the following relationships with RFa and RFb: 
 ω RFa =ω LO1 +ω LO2   (1)  ω HFb =ωLO 1 −ω LO2   (2)  Therefore, ω LO1= (ω RFa +ω RFb )/2  (3)  ω LO2 =(ω RFa −ωRFb)/2  (4)  
      In particular, if the first RF signal RFa is twice the second RF signal RFb, such as for the IEEE 802.11a in 5 GHz band and the IEEE 802.11b in 2.4 GHz band , a synthesizer  202  is provided in the embodiment to generate the first oscillation signal LO 1  and the second oscillation signal LO 2  from a common oscillation reference signal. The frequency of the oscillation reference signal VCO is chosen to be twice ω LO1 : 
 
ω VCO 2ω LO1 = 3/2ω RFa   (5) 
 
      Frequencies of the first oscillation signal LO 1  and second oscillation signal LO 2  can be determined to be as follows: 
 
ω LO1 =¾ω RFa   (6) 
 
ω LO2 =¼ω RFa   (7) 
 
      Based on the formulae, the first oscillation signal LO 1  can be derived by dividing the oscillation reference signal VCO by two, and the second oscillation signal LO 2  is obtained by dividing the first oscillation signal LO 1  by three. Thus, only one oscillation reference signal is required.  
       FIG. 3  shows an embodiment of synthesizer  202  in  FIG. 2   a . The synthesizer  202  comprises an oscillator  310 , providing an oscillation reference signal VCO having a frequency of 3/2 of the first RF signal RF a . A first divider  302  is coupled to the oscillator  310 , dividing the oscillation reference signal VCO by two, such that an inphase part of the first oscillation signal LO 1   I  and a quadrature part of the first oscillation signal LO 1   Q  are generated, having a frequency identical to ¾ of the first RF signal RF a . An inphase divider  304 I and a quadrature divider  304 Q are coupled to the first divider  302 , each dividing the inphase and quadrature parts of the first oscillation signal by three to generate inphase and quadrature parts of the second oscillation signal respectively. In this way, only one oscillation reference signal VCO is required to demodulate both 5 GHz and 2.4 GHz signals in a dual-band multi-mode 802.11 a/big transceiver.  
       FIG. 4   a  shows an embodiment of an RF demodulator  210  and an IF demodulator  212 . The RF demodulator  210  receives the inphase part of the first oscillation signal LO 1   I  to down convert the first RF signal RF a  and second RF signal RF b  to the intermediate signal IF. RF a  passes through the first low noise amplifier  402   a  and RF b  passes through the second low noise amplifier  402   b  respectively before coupling to the common RF mixer  404 . The RF mixer utilizes either the inphase part or the quadrature part of the first oscillation signal LO 1   I  to down-convert the first or second RF signals RF a  and RF b  to the intermediate signal IF.  
      For example, when down converting the first RF signal RF a , the following calculations are applicable. The first RF signal RFa can be expressed in the complex form:  
                       s   RFa     ⁡     (   t   )       =         I   ⁡     (   t   )       ⁢     cos   ⁡     (       ω   RFa     ⁢   t     )         -       Q   ⁡     (   t   )       ⁢     sin   ⁡     (       ω   RFa     ⁢   t     )                       =     Re   ⁡     (         R   BB     ⁡     (   t   )       ·     ⅇ       jω   RFa     ⁢   t         )                   =       1   2     ⁢     (           R   BB     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω   RFa     ⁢   t         +         R   BB   *     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω   RFa     ⁢   t           )                     (   8   )             
 
      Where I(t) and Q(t) are inphase and quadrature components of the baseband signal R BB (t) being transmitted, and R BB (t)=I(t)+j·Q(t). The first RF signal RF a  is down converted by the RF demodulator  210 , outputting the intermediate signal IF as:  
                       S   IF     ⁡     (   t   )       =       ⁢           s   RFa     ⁡     (   t   )       ⨯   LO     ⁢           ⁢     1   I     ⁢     (   t   )                   =       ⁢       1   2     ⁢       (           R   BB     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω   RFa     ⁢   t         +         R   BB   *     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω   RFa     ⁢   t           )     ·                       ⁢       1   2     ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       +     ⅇ       -     jω     LO   ⁢           ⁢   1         ⁢   t         )                   =       ⁢       1   4     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )                   =       ⁢       1   2     ⁢     (     Re   (           R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                           ⁢     Re   ⁡     (         R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         )       )                 (   9   )             
 
      The intermediate signal IF is then simultaneously sent to the inphase IF mixer  412   a  and the quadrature IF mixer  412   b . The outputs of the IF mixers  412   a  and  412   b  are inphase baseband mixed signal and quadrature baseband mixed signal respectively as explained below. In the inphase IF mixer  412   a , the intermediate signal IF is multiplied by the inphase part of the second oscillation signal LO 2   I :  
                       s   I     ⁡     (   t   )       =       ⁢         s   IF     ⁡     (   t   )       ·     cos   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                     =       ⁢       1   4     ⁢       (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )     ·     1   2       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t       +     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                   =       ⁢         1   8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )       +                     ⁢       1   8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢         1   8     ⁢     (         R   BB     ⁡     (   t   )       +       R   BB   *     ⁡     (   t   )         )       +     (     high   ⁢           ⁢   frequency   ⁢           ⁢   terms     )                   =       ⁢         I   ⁡     (   t   )       4     +     (     high   ⁢             ⁢             ⁢   frequency   ⁢           ⁢   terms     )                     (   10   )             
 
      Thus, through the first low-pass filter  414   a  and first variable gain amplifier  416   a , the high frequency terms are eliminated, and the I(t) is output as the inphase baseband signal.  
      Likewise, in the quadrature IF mixer  412   b , the intermediate signal IF is mixed with quadrature part of the second oscillation signal LO 2   Q , generating Q(t)/4 as shown below:  
      The second low-pass filter  414   b  and second variable gain amplifier  416   b  perform filtering and amplifying so that the Q(t) is amplified and output as the quadrature baseband signal.  
                       s   Q     ⁡     (   t   )       =       ⁢         s   IF     ⁡     (   t   )       ·     sin   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                     =       ⁢       1   4     ⁢       (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     RF   ⁢           ⁢   a       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )     ·       -   j     2       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t       -     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                   =       ⁢           -   j     8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )       +                     ⁢         -   j     8     ⁢     (               -       R   BB     ⁡     (   t   )         ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         -                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         +   -                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         -                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢           -   j     8     ⁢     (       -       R   BB     ⁡     (   t   )         +       R   BB   *     ⁡     (   t   )         )       +     (     high   ⁢           ⁢   frequency   ⁢             ⁢             ⁢   terms     )                   =       ⁢       -       Q   ⁡     (   t   )       4       +     (     high   ⁢             ⁢             ⁢   frequency   ⁢           ⁢   terms     )                     (   11   )             
 
      Similarly, for the case of second RF signal RF b  represented by:  
                       s   RFb     ⁡     (   t   )       =         I   ⁡     (   t   )       ⁢     cos   ⁡     (       ω   RFb     ⁢   t     )         -       Q   ⁡     (   t   )       ⁢     sin   ⁡     (       ω   RFb     ⁢   t     )                       =     Re   ⁡     (         R   BB     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω   RFb     ⁢   t         )                   =       1   2     ⁢     (           R   BB     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω   RFb     ⁢   t         +         R   BB   *     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω   RFb     ⁢   t           )                     (   12   )             
 
      The intermediate signal IF is obtained by multiplying the second RF signal RF b  by the inphase part of the first oscillation signal LO 1   I ,  
                       s   IF     ⁡     (   t   )       =       ⁢           s   RFb     ⁡     (   t   )       ⨯   LO     ⁢           ⁢     1   I     ⁢     (   t   )                   =       ⁢       1   2     ⁢       (           R   BB     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω   RFb     ⁢   t         +         R   BB   *     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω   RFb     ⁢   t           )     ·                       ⁢       1   2     ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       +     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t         )                   =       ⁢       1   4     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )                   =       ⁢       1   2     ⁢     (       Re   ⁡     (         R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1         )       ⁢   t         )       +                         ⁢     Re   ⁡     (         R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         )       )                 (   13   )             
 
      The inphase IF mixer  412   a  multiplies the intermediate signal IF with the inphase part of the second oscillation signal LO 2   I as:  
                       s   I     ⁡     (   t   )       =       ⁢         s   IF     ⁡     (   t   )       ·     sin   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                     =       ⁢       1   4     ⁢       (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     RF   ⁢           ⁢   b       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )     ·     1   2       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t       -     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                   =       ⁢         1   8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )       +                     ⁢       1   8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFa     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFa     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢         1   8     ⁢     (         R   BB     ⁡     (   t   )       +       R   BB   *     ⁡     (   t   )         )       +     (     high   ⁢           ⁢   frequency   ⁢             ⁢             ⁢   terms     )                   =       ⁢       -       I   ⁡     (   t   )       4       +     (     high   ⁢             ⁢             ⁢   frequency   ⁢           ⁢   terms     )                     (   14   )             
 
      After passing through the first low-pass filter  414   a  and first variable gain amplifier  416   a , only I(t) components are output as the inphase baseband signal. Likewise, the quadrature IF mixer  412   b  multiplies the intermediate signal IF by the quadrature part of the second oscillation signal LO 2   Q  to generate the quadrature baseband signal:  
                       s   Q     ⁡     (   t   )       =       ⁢         s   IF     ⁡     (   t   )       ·     sin   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                     =       ⁢       1   4     ⁢       (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     RF   ⁢           ⁢   b       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )     ·       -   j     2       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t       -     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                   =       ⁢           -   j     8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )       +                     ⁢         -   j     8     ⁢     (                 R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         -                     R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         +   -                     R   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω   RFb     -     ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         -                   R   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω   RFb     +     ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢           -   j     8     ⁢     (         R   BB     ⁡     (   t   )       +       R   BB   *     ⁡     (   t   )         )       +     (     high   ⁢           ⁢   frequency   ⁢             ⁢             ⁢   terms     )                   =       ⁢         Q   ⁡     (   t   )       4     +     (     high   ⁢             ⁢             ⁢   frequency   ⁢           ⁢   terms     )                     (   15   )             
 
      The second low-pass filter  414   b  and second variable gain amplifier  416   b  amplify the output from quadrature IF mixer  412   b  to output the Q(t) as the quadrature baseband signal. As described, through carefully chosen first oscillation signal LO 1  and second oscillation signal LO 2 , the first and second RF signals RF a  and RF b  can be down-converted with common RF and IF mixers.  
       FIG. 4   b  shows an embodiment of a RF modulator  220  and an IF modulator  222 . The inphase baseband transmit signal I(t) and quadrature baseband transmit signal Q(t) are modulated by the first inphase mixer  426   a , second inphase mixer  426   b  and subtractor  430  to form the intermediate signal IF of the form:  
                       g   IF     ⁡     (   t   )       =         I   ⁡     (   t   )       ⁢     cos   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )         -       Q   ⁡     (   t   )       ⁢     sin   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                       =     Re   ⁡     (         g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                   =       1   2     ⁢     (           g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )                     (   16   )               
      wherein the signal I(t)cos(ω LO2 t) is a first preliminary intermediate signal, and the signal Q(t) sin(ω LO2 t) is a second preliminary intermediate signal.  
      The RF mixer  424  multiplies the intermediate signal IF by either the inphase or the quadrature part of the first oscillation signal LO 1   I , to generate a signal comprising both bands. While multiplying the inphase part of the first oscillation signal LO 1 :  
                       g   RF     ⁡     (   t   )       =       ⁢         g   IF     ⁡     (   t   )       ⁢   LO   ⁢           ⁢     1   I     ⁢     (   t   )                   =       ⁢         g   IF     ⁡     (   t   )       ⁢     cos   ⁡     (       ω     LO   ⁢           ⁢   1       ⁢   t     )                     =       ⁢       1   2     ⁢       (           g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )     ⨯                       ⁢       1   2     ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       +     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t         )                   =       ⁢       1   4     ⁢     (                 g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                     g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )         ⁢   t         +                     g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +                   g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )         ⁢   t               )                   =       ⁢         1   4     ⁢     (           g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )         ⁢   t           )       +                     ⁢       1   4     ⁢     (           g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )       ⁢   t         +         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )         ⁢   t           )                   =       ⁢         1   2     ⁢     Re   ⁡     (         g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω   RFa     ⁢   t         )         +       1   2     ⁢     Re   ⁡     (         g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω   Rfb     ⁢   t         )                       =       ⁢         1   2     ⁢     (         I   ⁡     (   t   )       ⁢   cos   ⁢           ⁢     ω   RFa     ⁢   t     -       Q   ⁡     (   t   )       ⁢   sin   ⁢           ⁢     ω   RFa     ⁢   t       )       +                     ⁢       1   2     ⁢     (         I   ⁡     (   t   )       ⁢   cos   ⁢           ⁢     ω   Rfb     ⁢   t     -       Q   ⁡     (   t   )       ⁢   sin   ⁢           ⁢     ω   RFb     ⁢   t       )                     (   17   )             
 
      A selection mechanism may be provided to filter the output from RF mixer  424 . For first RF signal RF a , components with frequency equal to the sum of first oscillation signal LO 1  and second oscillation signal LO 2  are selected and output after amplification by the first power amplifier  422   a :  
                 g   RFa     ⁡     (   t   )       =       1   2     ⁢     (         I   ⁡     (   t   )       ⁢   cos   ⁢           ⁢     ω   RFa     ⁢   t     -       Q   ⁡     (   t   )       ⁢   sin   ⁢           ⁢     ω     RF   ⁢           ⁢   a       ⁢   t       )               (   18   )             
 
      For second RF signal RF b , components with frequency equal to the difference of first oscillation signal LO 1  and second oscillation signal LO 2  are selected and output after amplification by the second power amplifier  422   b :  
                 g   RFb     ⁡     (   t   )       =       1   2     ⁢     (         I   ⁡     (   t   )       ⁢   cos   ⁢           ⁢     ω   RFb     ⁢   t     -       Q   ⁡     (   t   )       ⁢   sin   ⁢           ⁢     ω   RFb     ⁢   t       )               (   19   )             
 
      Likewise. while multiplying the quadrature part of the first oscillation signal LO 1 :  
                       g   RF     ⁡     (   t   )       =       ⁢         g   IF     ⁡     (   t   )       ⁢   LO   ⁢           ⁢     1   Q     ⁢     (   t   )                   =       ⁢         g   IF     ⁡     (   t   )       ⁢     sin   ⁡     (       ω     LO   ⁢           ⁢   1       ⁢   t     )                     =       ⁢       1   2     ⁢       (           g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ⁢     ⅇ     0   ⁢   j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )     ⨯                       ⁢       1     2   ⁢   j       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       -     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t         )                   =       ⁢       1     4   ⁢   j       ⁢     (                 g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         +         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )         ⁢   t         -                     g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )       ⁢   t         -         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )         ⁢   t                 )                   =       ⁢         1     4   ⁢   j       ⁢     (           g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )       ⁢   t         -         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       +     ω     LO   ⁢           ⁢   1         )         ⁢   t           )       -                     ⁢       1     4   ⁢   j       ⁢     (           g   BB     ⁡     (   t   )       ⁢     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )       ⁢   t         -         g   BB   *     ⁡     (   t   )       ⁢     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   2       -     ω     LO   ⁢           ⁢   1         )         ⁢   t           )                   =       ⁢         1   2     ⁢     Im   (         g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω     RF   ⁢           ⁢   a       ⁢   t         )       -       1   2     ⁢     Im   ⁡     (         g   BB     ⁡     (   t   )       ⁢     ⅇ     j   ⁢           ⁢     ω   RFb     ⁢   t         )                       =       ⁢         1   2     ⁢     (         I   ⁡     (   t   )       ⁢   sin   ⁢           ⁢     ω   RFa     ⁢   t     -       Q   ⁡     (   t   )       ⁢   cos   ⁢           ⁢     ω   RFa     ⁢   t       )       -                     ⁢       1   2     ⁢     (         I   ⁡     (   t   )       ⁢   sin   ⁢           ⁢     ω   RFb     ⁢   t     -       Q   ⁡     (   t   )       ⁢   cos   ⁢           ⁢     ω   RFb     ⁢   t       )                   =       ⁢         g   RFa     ⁡     (   t   )       -       g   RFb     ⁡     (   t   )                       (   20   )             
 
      In this way, hardware sharing is maximized using the proposed transceiver structure because RF signals from both frequency bands, i.e. the first RF signal RF a  and the second RF signal RF b , share the same path in transmitter and receiver.  
      To enhance sideband suppression in the transmitter, single sideband mixers can be used. This is possible because the inphase and quadrature components of the first oscillation signal LO 1  and the second oscillation signal LO 2  can be conveniently generated from the divide-by-two circuits in frequency synthesizer as illustrated in  FIG. 3 .  
       FIG. 5   a  shows another embodiment of the RF modulator  220  and IF modulator  222  using three single-sideband mixers to enhance sideband suppression. Modifications are made to the transmitter architecture as follows. A baseband transmit signal to be transmitted, comprising inphase signal I(t) and quadrature signal Q(t), is defined as: 
   g   BB ( t )= I ( t )+ jQ ( t )  (21)  
      Through modulation performed by the first IF mixer  520   a  with inphase LO 2  signal, the second IF mixer  520   b  with quadratre LO 2  signal, and the first adder  512   a , the first preliminary intermediate signal is generated as:  
                     g     IF   ⁢           ⁢   1   ⁢     (   t   )         =         I   ⁡     (   t   )       ⁢     cos   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )         +       Q   ⁡     (   t   )       ⁢     sin   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                       =     Re   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                   =       1   2     ⁢     (           g   BB     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )                     (   22   )             
 
      Similarly, the third IF mixer  520   c  with inphase LO 2  signal, the fourth IF mixer  520   d  with quadrature LO 2  signal, and the subtractor  512   b  generate the second preliminary intermediate signal as:  
                       g     IF   ⁢           ⁢   2       ⁡     (   t   )       =         I   ⁡     (   t   )       ⁢     sin   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )         -       Q   ⁡     (   t   )       ⁢     cos   ⁡     (       ω     LO   ⁢           ⁢   2       ⁢   t     )                       =     -     Im   ⁡     (         g   BB     ⁡     (   t   )       ⁣     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         )                     =       1     2   ⁢   j       ⁢     (         -       g   BB     ⁡     (   t   )         ·     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )                     (   23   )             
 
      The first RF mixer  504   a  then multiplies the first preliminary intermediate signal with the quadrature part of the first oscillation signal LO 1   Q :  
                       g     RF   ⁢           ⁢   1       ⁡     (   t   )       =       ⁢           g     IF   ⁢           ⁢   1       ⁡     (   t   )       ·   LO     ⁢           ⁢     1   Q     ⁢     (   t   )                   =       ⁢       1   2     ⁢       (           g   BB     ⁡     (   t   )       ·     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )     ·                       ⁢       1     2   ⁢   j       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       -     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t         )                   =       ⁢       1     4   ⁢   j       ⁢     (                 g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                     g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         -                     g   BB     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         -                   g   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢       1   2     ⁢     (       Im   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       +                         ⁢     Im   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ     j   (       ω     LO   ⁢           ⁢   1       +       ω     LO   ⁢           ⁢   2       ⁢   t             )       )                 (   24   )             
 
      The second RF mixer  504   b  multiplies the inphase part of the first oscillation signal LO 1   I , and second preliminary intermediate signal:  
                       g     RF   ⁢           ⁢   2       ⁡     (   t   )       =       ⁢           g     IF   ⁢           ⁢   2       ⁡     (   t   )       ·   LO     ⁢           ⁢     1   I     ⁢     (   t   )                   =       ⁢       1     2   ⁢   j       ⁢       (         -       g   BB     ⁡     (   t   )         ·     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )     ·                       ⁢       1   2     ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       +     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t         )                   =       ⁢       1     4   ⁢   j       ⁢     (               -       g   BB     ⁡     (   t   )         ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t                       -       g   BB     ⁡     (   t   )         ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         +                   g   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢       1   2     ⁢     (       -     Im   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )         +                         ⁢     Im   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )                 (   25   )             
 
      The combining unit  502  can generate the second RF signal RF b  by subtracting formula (21) from formula (22):  
                       g   RF     ⁡     (   t   )       =       ⁢         g     RF   ⁢           ⁢   1       ⁡     (   t   )       -       g     RF   ⁢           ⁢   2       ⁡     (   t   )                     =       ⁢       1   2     ⁢     (       Im   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       +                         ⁢     Im   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )               =       ⁢     Im   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )                     (   26   )             
 
      Therefore, the baseband transmit signal can be up-converted to the lower-band RF signals RF b  in this way. On the other hand, the higher-band RF signal RF a  can be generated in three different ways as described below.  
      First, the combining unit  502  can generate the first RF signal RF a  by adding formula (21) and formula (22):  
                       g   RF     ⁡     (   t   )       =       ⁢         g     RF   ⁢           ⁢   1       ⁡     (   t   )       +       g     RF   ⁢           ⁢   2       ⁡     (   t   )                     =       ⁢       1   2     ⁢     (       Im   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       +                           ⁢     Im   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )     +                   ⁢       1   2     ⁢     (       -     Im   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )         +                         ⁢     Im   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )               =       ⁢     Im   (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t                         (   27   )             
 
      Second, instead of performing the opposite adding and subtracting operations for the combining unit  502 , the IF outputs of adder  512   a  and subtractor  512   b  can be swapped to go into the RF mixers  504   a  and  504   b .  FIG. 5   b  shows a further embodiment of the RF modulator  220  and IF modulator  222  to demonstrate this approach. The output of adder  512   a  is sent to the second RF mixer  504   b  while the output of subtractor  512   b  is sent to the first RF mixer  504   a . Thus, the first RF mixer  504   a  up converts the second preliminary intermediate signal by the quadrature part of the first oscillation signal LO 1   Q :  
                       g     RF   ⁢           ⁢   1       ⁡     (   t   )       =       ⁢           g     IF   ⁢           ⁢   2       ⁡     (   t   )       ·   LO     ⁢           ⁢     1   Q     ⁢     (   t   )                   =       ⁢       1     2   ⁢   j       ⁢       (         -       g   BB     ⁡     (   t   )         ·     ⅇ       -   j     ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   2       ⁢   t           )     ·                       ⁢       1     2   ⁢   j       ⁢     (       ⅇ     j   ⁢           ⁢     ω     LO   ⁢           ⁢   1       ⁢   t       -     ⅇ       -     jω     LO   ⁢           ⁢   1         ⁢   t         )                   =       ⁢       -     1   4       ⁢     (               -       g   BB     ⁡     (   t   )         ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +                   g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t                         g   BB     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         -                   g   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t               )                   =       ⁢       1   2     ⁢     (       Re   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       -                         ⁢     Re   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )                 (   28   )             
 
      The second RF mixer  504   b  performs the other up conversion likewise:  
                       g     RF   ⁢           ⁢   2       ⁡     (   t   )       =       ⁢           g     IF   ⁢           ⁢   1       ⁡     (   t   )       ·   LO     ⁢           ⁢     1   I     ⁢     (   t   )                   =       ⁢       1   2     ⁢       (           g   BB     ⁡     (   t   )       ·     ⅇ       -     jω     LO   ⁢           ⁢   2         ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ       jω     LO   ⁢           ⁢   2       ⁢   t           )     ·     1   2       ⁢     (       ⅇ       jω     LO   ⁢           ⁢   1       ⁢   t       +     ⅇ       -     jω     LO   ⁢           ⁢   1         ⁢   t         )                   =       ⁢       1   4     ⁢     (                 g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t                           g   BB     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )         ⁢   t         +         g   BB   *     ⁡     (   t   )       ·     ⅇ       -     j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )         ⁢   t                 )                   =       ⁢       1   2     ⁢     (       Re   ⁡     (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       +     Re   ⁡     (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )         )                     (   29   )             
 
      The combining unit  502  then still performs a subtraction of equation (28) by equation (29) to generate first RF signal RF a :  
                       g   RF     ⁡     (   t   )       =       ⁢         g     RF   ⁢           ⁢   1       ⁡     (   t   )       -       g     RF   ⁢           ⁢   2       ⁡     (   t   )                     =       ⁢         1   2     ⁢     (       Re   (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )     -     Re   (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )       -                     ⁢       1   2     ⁢     (       Re   (         g   BB     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       -     ω     LO   ⁢           ⁢   2         )       ⁢   t         )     +     Re   (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )       )                   =       ⁢     -     Re   (         g   BB   *     ⁡     (   t   )       ·     ⅇ       j   ⁡     (       ω     LO   ⁢           ⁢   1       +     ω     LO   ⁢           ⁢   2         )       ⁢   t         )                     (   30   )             
 
      Transmit baseband signals I(t) and Q(t) are therefore up-converted to higher frequency first RF signal RF a .  
      For the third way to generate RF a , if the inphase and quadrature parts of the first oscillation signal sent to the first RF mixer  504   a  and second RF mixer  504   b  in  FIG. 5   a  are swapped, the same result can be derived as shown in the formulae (29) and (30).  
      In summary, a sliding IF dual-band transceiver architecture is proposed. The usage of sum and difference of the first and second oscillation signals LO 1  and LO 2  effectively maximizes hardware sharing by allowing the signals for two different frequency bands to pass through the same signal path. When the first RF is twice the second RF, only one frequency synthesizer is required to generate local oscillation signals for the two frequency conversion stages and the two frequency bands Additionally, the transmitter architecture utilizing single sideband mixers enhances sideband suppression.  
      While the invention has been described by way of example and in terms of preferred embodiment, it is to be understood that the invention is not limited thereto. To the contrary, it is intended to cover various modifications and similar arrangements (as would be apparent to those skilled in the art). Therefore, the scope of the appended claims should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements.