Abstract:
Simplifying functions representing raised sine or cosine curves to functions representing simple sine or cosine curves makes it possible to implement an electrical equivalent circuit for a ramp generator. The core of the ramp generator with an output power level controller is second-order direct-form feedback structure ( 60 ), which forms a digital sinusoidal oscillator. The initial values of two state variables x2(n), x2(n+1) of the oscillator are chosen so that they both contain a predetermined first constant value. This first constant value will emerge as the amplitude value of the pure sine wave generated by the oscillator. Particularly the first constant value is equal to the desired nominal level A of the ramp minus the starting level. A second constant value (A+dc) is added to the oscillator output. The added result is scaled ( 66 ) so that the nominal power level is A. A multiplexer ( 67 ) keeps the power level between the ramps constant.

Description:
This application is a continuation in part of PCT/F100/01065, filed Dec. 1, 2000, which claims priority from Finnish Serial Number 19992614, filed Dec. 3, 1999. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to a transmitter of a radio system using burst transmission, and particularly to a ramp generator for shaping the rise and the fall of a burst and for controlling the power level of the burst. 
     BACKGROUND OF THE INVENTION 
     A burst is the transmission quantum of numerous digital radio systems based on the principle of time division duplex (TDD), frequency division duplex (FDD), and code division duplex (CDD). The transmission takes place during a short time window. Within this time interval the emission rises from the starting power level to the nominal power level. The signal is then modulated to transmit a packet of bits. After that, the power level decreases until it reaches the minimum power level. The time mask of the burst, during which the bits are transmitted, is called a useful part or a payload part. Modulation is performed in the transmitter analogically or digitally, either at a base band frequency or at an intermediate frequency (IF). The modulated IF signal is then mixed up to the radio transmission frequency. 
     FIG. 1 depicts the rising portion of the power envelope of a burst comprising a digitally modulated intermediate frequency signal. In this example, during the rise of the burst the envelope should track a raised sine curve. After a predetermined period the power has reached its selected nominal level, whereupon modulation starts. This instant is denoted as 0 in the horizontal time axis. Usually the transmission power is adjustable according to the requirement of the system concerned. For that reason the nominal power level value can vary between the maximum power level and the minimum power level, as shown by the dotted line and the dashed line in FIG.  1 . The nominal power level between those levels can usually attain one of several discrete power levels. For example, the downlink dynamic power control in the GSM system uses 16 power levels with 2 dB separations. 
     FIG. 2 depicts the falling portion of the envelope of a digitally modulated intermediate frequency signal. The signal envelope during the fall of the transmission burst should track a raised cosine curve. 
     The power level can be controlled burst by burst. Control is realized by scaling the ramp curve which follows the raised sine/cosine curve. Hence, the ramp-up curve starts from the minimum power level, but settles at the level specified by the power level indication as shown in FIG.  1 . At the end of the burst the ramp-down curve starts from the nominal power level, but settles at the minimum power level as shown in FIG.  2 . 
     Conventionally, power ramping and control of the output power level are performed in the analog domain. One problem with analog solutions is inaccuracies caused by aging and by variations in operation temperature and components. Furthermore, the analog solutions are complex, and stability is a problem. 
     Today the tendency is to perform power ramping and output power level control digitally in order to avoid the afore-mentioned problems. Some basic solutions are presented below. 
     FIG. 3 illustrates in broad outline the formation of a modulated IF signal into a shape as shown in FIG.  1  and FIG.  2 . Data symbols arrive at digital modulator  31  that carries out modulation according to the modulation scheme of the system concerned. A ramp generator in block  32  generates the rising and falling edges according to the raised sine/cosine curve and a flat portion between the curves. Digital output signals from the ramp generator and the modulator are then converted to analog signals in digital-to-analog converters  33  and  36 : For removing the high frequency sampling components the analog signals are then filtered in low pass filters  37  and  38 , whereupon the analog modulated signal is multiplied in analog multiplier  35  by the analog ramp signal in order to smoothen out the rise and fall of the burst. The output from the multiplier is the analog modulated IF signal with ramped power. 
     FIG. 4 illustrates another digital modulator. Data symbols arrive at digital modulator  41 , which carries out modulation according to the modulation scheme of the system concerned and produces I and Q signals. Said signals are then converted into analog signals by DA converters  43  and  44 . For removing the high frequency sampling components the analog signals are filtered in low pass filters  410  and  411 , whereupon both the analog I signal and the analog Q signal are transformed into an intermediate frequency by mixer  45  and mixer  46 , accordingly. After mixing the sum of I signal and Q signal are added up in analog adder  47  to form the sum signal. A ramp generator in block  42  generates the ramp signal, i.e. the rising and falling edges according to the raised sine/cosine curve and the flat portion between them. The digital ramp signal is then converted into an analog signal in converter  48 . The high frequency sampling components are filtered in low pass filter  412 , whereupon the analog modulated sum signal is multiplied in multiplier  49  by the analog ramp signal in order to smooth out the rise and fall of the IF burst. The output from the multiplier is the analog modulated IF signal. 
     Common to both prior art solutions described above are the performance of both the modulation and the generation of the ramp signal digitally but conversion of the digital result signals into the analog domain before multiplying. However, there is a tendency in the art to carry out all processes within the digital domain. In order to better understand one possible realization of the digital ramp generator, a short review of the mathematical background is of assistance. 
     The burst signal can be considered as a product of an original modulated signal m(t) and a periodical switching signal sw(t). The spectrum of the burst signal is the convolution of the spectra of these two signals in the frequency domain. 
     For rectangular switching, i.e. without raised cosine/sine shaping, formula (1) is valid for frequency response:                W        (   f   )       =         M        (     f   -     f   C       )       *     Sw        (   f   )         =     K          ∑     n   =     -   ∞       ∞                       M        (     f   -     f   C     -     nf   g       )          sin                 π                   nf   g          τ     π                   nf   g        τ                       (   1   )                                
     where 
     * denotes convolution, 
     f c  is the carrier frequency, 
     f g  is the burst gating rate, 
     τ is the burst length and 
     K is a proportional constant. 
     For raised cosine/sine switching, i.e. with raised cosine/sine shaping, formula (2) is valid for frequency response:                W        (   f   )       =     H          ∑     n   =     -   ∞       ∞                       M        (     f   -     f   C     -     nf   g       )            (     τ   -     T   r       )        sin                   c        (       nf   g          (     τ   -     T   r       )       )              cos        (     π                   T   r          nf   g       )         1   -       (     2        T   r          nf   g       )     2                       (   2   )                                
     where T r  indicates the ramp duration, and H is proportional constant. 
     The spectrum of the periodic burst signal consists of infinite numbers of secondary spectral lobes having the same shape as M(f) separated by the burst gating rate f g , and having decreasing amplitudes. Since the secondary spectral lobes resulting from formula (2) decay faster than those resulting from formula (1), the raised cosine/sine switching is used. 
     The following function is used to smooth out the rise of the burst:                    (     A   -   dc     )            sin        (       π                 t       2        T   r         )       2       +              dc     ,           (   3   )                                
     where 
     T r  indicates the ramp duration, 
     t is [0 T r ], 
     A is the envelope of the modulated signal, and 
     dc is the dc offset which settles the starting power level in FIG.  1 . 
     The following function (4) is used to smooth out the fall of the burst:                  (     A   -   dc     )            cos        (       π                 t       2        T   r         )       2       +                dc   .             (   4   )                                
     FIG. 5 illustrates a ramp generator and an output power controller known in the art which are based on formulas (3) and (4). The raised sine values of formula (3) or the raised cosine values of formula (4) are stored in the read only memory (ROM)  51 . Digital multiplier  52  is used to control the amplitude level, i.e. value (A−dc). Adder  53  sets the dc offset, i.e. the last factor of formulas. The size of the ROM memory is about (fcfk×T r )×outw, where f clk  is the digital IF modulator clock frequency (sampling frequency), T r  is the pulse duration and outw is the multiplier input width. 
     One drawback of this known ramp generator is that due to the high clock frequency in the digital IF modulators, the size of the memory is large. For example, if the clock frequency is 52 MHz and the ramp duration is 14 μs, then the size of the memory is about 728×12 bit. Furthermore, a multiplier is needed as an extra component to set the output power level. 
     Another possible way to implement the ramp generator and output power controller is to use a FlR-filter (Finite Impulse Response). The number of the FIR filter taps is f clk ×T r , where f clk  is the digital IF modulator clock frequency and T r  is the pulse duration. One drawback of filter implementation is that due to the high clock frequency (sampling frequency) in the IF modulators, there are many taps in the FIR. Therefore, the realization of raised sine and cosine functions with filters is complex. For example, with the above mentioned values, i.e. the clock frequency is 52 MHz and the ramp duration is 14 μs, the number of the FIR filter taps is 728. 
     SUMMARY OF THE INVENTION 
     One objective of the present invention is to devise a digital ramp generator with an output power controller that is easy to implement and which requires a minimum number of standard components, without the need for raised sine and cosine memories or digital filters. 
     A further objective is to devise a digital ramp generator with an output power controller generating a digital output signal that can directly multiply the digital modulated signal produced by a digital modulator. 
     Yet a further objective is to devise a digital ramp with inherent power control, wherein the generator and power control form a functionally inseparable integrated unit. 
     The present invention is based on a further mathematical explication of the raised sine and cosine curves representing the rise and fall of the burst. Simplifying functions representing raised sine/cosine curves to functions representing simple sine/cosine curves makes it possible to implement an electrically equivalent circuit consisting of a few simple basic components while tracking the raised sine and cosine functions well. 
     The core of the electrical equivalent circuit functioning as a ramp generator with an output power level controller is a second-order direct-form feedback structure forming a digital sinusoidal oscillator. The structure is well-known as such, and it produces an output sequence which is the sampled version of the pure sine wave with an amplitude value. The initial values of two state variables of the oscillator are chosen so that they both contain a predetermined first constant value. This first constant value will emerge as the amplitude value of the pure sine wave generated by the oscillator. Particularly the first constant value is equal to the desired nominal level A of the ramp minus a dc offset, where the dc determines the starting power level of the rising ramp and the settling power level after the falling ramp. The dc offset may also be called as a base level. 
     A second constant value equal to the desired nominal level of the ramp plus the above-mentioned dc offset is added to the oscillator output. Due to the deliberately chosen first and second constant values, the adding operation causes the rising ramp to start from level 2·dc and to end at level 2·A. Accordingly, the adding operation causes the falling ramp to start from level 2·A and to end at level 2·dc. 
     Finally, the result will be scaled so that the nominal power level will be A, and the starting level of the rising ramp and the end level of the falling ramp will be dc. 
     After the ramp has risen to the predetermined nominal power level, the output power level will be kept constant up to the instant when the ramp starts falling. 
     The proposed digital ramp generator and power controller can be implemented with the aid of two two-input adders, two delays, a multiplexer, and a fixed multiplier, which can be constructed with (N−1) adders, where N is the number of non-zero bits in the coefficient. The proposed ramp generator and output power controller saves hardware compared to the conventional methods. Furthermore, since the proposed ramp generator and output power level controller needs neither a memory nor a multiplier, it can be easily implemented with standard cells. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention is described more closely with reference to the accompanying drawings, in which 
     FIG. 1 shows the rising ramp of a burst; 
     FIG. 2 shows the falling ramp of a burst; 
     FIG. 3 illustrates a prior art modulator and ramp generator; 
     FIG. 4 depicts another prior art modulator and ramp generator; 
     FIG. 5 shows a prior art digital ramp generator; 
     FIG. 6 is a block diagram of the invented ramp generator; 
     FIG. 7 is another embodiment of the invented ramp generator; 
     FIG. 8A shows output error when truncation is used; 
     FIG. 8B shows output error when rounding is used; 
     FIG. 9 depicts a ramp down profile; 
     FIG. 10 depicts a ramp up profile; 
     FIG. 11 shows a ramp generator adapted to Blackman window, and 
     FIG. 12 depicts a block diagram of another ramp generator adapted to Blackman window. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The mathematical background of the invented ramp generator is founded on further development of known formulas for raised cosine and sine functions. 
     Using trigonometric identities, the previously presented function (3), which is used for smoothing out the rise of the burst, can be rearranged as follows:                1   2          (       (     A   +     d                 c       )     +       (     A   -     d                 c       )          cos        (         π                 t       T   r       +   π     )           )             (   5   )                                
     At the time instant t=0 the value of the function is dc, and at the end of ramp. time T r  the value of the function is A. Hence, dc is the offset value which settles the starting power level, and A is the power level which the ramp reaches at the end (see FIG.  1 ). Therefore, the setting of the output power level of the ramp generator, i.e. the nominal power level of the burst, can be done by controlling the value of A. It is essential to note that in equation (5) the cosine term is not raised, so it can be implemented by a sinusoidal oscillator. 
     Accordingly, using trigonometric identities, function (4), which is used for smoothing out the fall of the burst, can be rearranged as follows:                  1   2          (       (     A   +   dc     )     +       (     A   -   dc     )          cos        (       π                 t       T   r       )           )       ,           (   6   )                                
     At the time instant t=0 when the ramp begins to fall, the value of the function is A, and at the end of ramp time T r  the value of the function is dc. Hence, dc is the offset value which settles the power level after the ramp, and A is the power level from which the ramp begins to fall (see FIG.  1 ). Also in this equation the cosine term is not raised, so it can be implemented by a sinusoidal oscillator. 
     FIG. 6 depicts a ramp generator with an output power level controller which realizes the above-mentioned formulas (5) and (6). The circuit consists of the digital sinusoidal oscillator  60 , adder  68 , scaler  66 , and MUX  67 . 
     The core of this structure, the digital sinusoidal oscillator  60 , is a second-order direct-form feedback structure known as such. It produces the cosine term of equations (5) and (6) and amplitude A−dc of the cosine term. Adder  68  adds the constant value A+dc to the oscillator output. 
     Operation of the oscillator will now be described in more detail by explaining the mathematical basis of the operation of the oscillator with reference to the circuit elements in FIG.  6 . 
     Digital sinusoidal oscillator  60  producing the cosine term of formulas (5) and (6) is implemented by the following second-order difference equation: 
     
       
           x   2 ( n+ 2)= αx   2 ( n+ 1)− x   2 ( n ).  (7) 
       
     
     FIG. 6 shows the signal flow graph of the second-order direct-form feedback structure with state variables x 1 (n) and x 2 (n). State variable x 1 (n) is the input to delay block  62 , and state variable x 2 (n) is the output from that block. Delay block  62  can be implemented with registers. As shown, state variable x 1 (n) is also applied to block  63 , which multiplies it by the coefficient α, whereas state variable x 2 (n) is applied to block  64 , which takes negation from input signal, i.e. multiplies it by the coefficient −1. The outputs from blocks  63  and  64  are fed to the two-input adder  65 . The added result, which is the right-hand part of equation 7, is then applied to delay block  61 , which can be implemented with registers. The added result is denoted as x 2 (n+2) in FIG.  6 . 
     As shown in the figure, the two state variables are related by the equation: 
     
       
           x   1 ( n )= x   2 ( n +1)  (8) 
       
     
     Solving the one-sided z transform of equation (7) for x 2 (n) leads to formula                    X   2          (   z   )       =           (       z   2     -     α                 z       )            x   2          (   0   )         +     z                     x   1          (   0   )               z   2     -     α                 z     +   1         ,           (   9   )                                
     where x 1 (0) is the initial value of state variable x 1 (n), i.e. the input to delay block  62 , and x 2 (0) is the initial value of state variable x 2 (n), i.e. the output from delay block  62 . 
     Identifying the second state variable as output variable 
     
       
           y ( n )= x   2 ( n ),  (10) 
       
     
     as shown in FIG. 6, and choosing the denominator coefficient α to be 
     
       
         α=2 cos θ 0 , θ 0 =ω 0    T= 2πƒ 0 /ƒ clk ,  (11) 
       
     
     where f 0  is the oscillator frequency and f clk  is the sampling frequency, and choosing the initial values of the state variables to be 
     
       
           x   1 (0)= A *cos θ 0   , x   2 (0)= A* , where  A*=A−dc   (12) 
       
     
     we obtain from equation (9) a discrete-time sinusoidal function as the output signal:                Y        (   z   )       =       A   *     (       z   2     -     cos                   θ   0        z       )           z   2     -     2      cos                   θ   0        z     +   1               (   13   )                                
     This output signal Y(z) has complex-conjugate poles at 
     
       
           z= exp(± jθ   0 ),  (14) 
       
     
     and a unit sample response 
     
       
           y ( n )= A* cos( nθ   0 ),  n≧ 0  (15) 
       
     
     Thus the impulse response of the second-order system with complex-conjugate poles on the unit circle is a sinusoidal waveform. 
     An arbitrary initial offset φ 0  can be realized, namely, 
     
       
           y ( n )= A *cos(θ 0   n+φ   0 )  (16) 
       
     
     by choosing the initial values: 
     
       
           x   1 (0)= A *cos(θ 0 φ 0 )  (17) 
       
     
     
       
           x   2 (0)= A *cos(φ 0 ).  (18) 
       
     
     The output frequency of the digital oscillator (θ 0 ) could be altered by changing the coefficient α in (11) and the initial value in (17). The above derived formulas show that any real-valued sinusoidal oscillator signal can be generated by the second-order structure shown in FIG.  6 . The initial phase offsets of the digital oscillator are 0 for the ramp down and π for the ramp up. The initial values of state variables x 1 (n) and x 2 (n) for these phase offsets are calculated from equations 17 and 18. 
     Hence, for the failing ramp (φ 0 =0) the initial values are 
     x 1 (0)=(A−dc)cos(θ 0 ) 
     x 2 (0)=(A−dc). 
     For the rising ramp (φ 0 =π) the initial values are 
     x 1 (0)=−(A−dc)cos(θ 0 ) 
     x 2 (0)=−(A−dc). 
     The initial values for the raising ramp are negation of the initial values for the falling ramp. 
     The output sequence y(n) of the ideal oscillator is the sampled version of a pure sine wave. The angle θ 0  represented by the oscillator coefficient is given by 
     
       
         θ 0 =2π ƒ   0   T,   (19) 
       
     
     where f 0  is the desired frequency in cycles per second. In actual implementation, the multiplier coefficient 2 cos θ 0  is assumed to have b+2 bits. In particular, one bit is for the sign, one bit for the integer part, and b bits for the remaining fractional part in fixed-point number representation. Then the largest value of the coefficient 2 cos θ 0  which can be represented is (2−2 −b ). This value of the coefficient gives the smallest value of θ min  which can be implemented by a direct form digital oscillator using b bits:                θ     m                 i                 n       =         cos     -   1            [       1   2          (     2   -     2     -   b         )       ]       .             (   20   )                                
     Therefore, the smallest frequency which the oscillator can generate is                  f     m                 i                 n       =         θ     m                 i                 n         2      π            f   clk         ,           (   21   )                                
     where f clk  is the clock frequency (sampling frequency). 
     As an example, let b=25 bits. The largest oscillator coefficient (2 cos θ 0 ) is 67108863/33554432, then θ min =cos −1 (67108863/67108864)≈0.00017263. If clock frequency f clk  is 52 MHz and b=25, then f min ≈1.43 kHz. 
     The ramp duration could be altered by changing the output frequency of the digital oscillator. During the ramp period the phase change in equations 5 and 6 is π, and therefore the required output frequency is                f   0     =       1     2        T   r         .             (   22   )                                
     The smallest frequency f min  should be below f 0 . For T r 32 14 μs, f 0 ≈35.71 kHz. 
     Referring to FIG. 6 the cosine term in formulas 5 and 6 is implemented by the sinusoidal oscillator, which solves the second order difference equation. Value A* of the digital oscillator amplitude is selected to be same as A−dc, where A is the desired nominal power level of the ramp signal and dc is the level from which the rising ramp starts and to which the falling ramp ends. The initial phase offsets of the digital oscillator are 0 when forming the ramp down, and π when forming the ramp up. Hence, it is worth noting, that by choosing the value A−dc from a table of values the desired power level of the digital ramp signal is selected. By setting the initial phase offset of the digital oscillator to 0, the oscillator generates the ramp down signal, and by setting the initial phase offset to π, the oscillator generates the ramp up signal. 
     The digital output signal of the oscillator is (A−dc)cos(nθ 0 +φ 0 ). However, in order to fulfill requirements of formulas 5 and 6, a factor A+dc should be added up said digital output signal. That&#39;s why the output signal and a signal having the value A+dc are applied to the two-input adder  68  (see FIG.  6 ). By adding, a ramp signal is produced with an amplitude which is twice as high as required. For that reason the ramp signal is applied to scaler  66 , which performs binary shift 2 −1  to the incoming digital ramp signal. The scaler can be implemented with wiring. 
     Finally, the digital ramp signal is fed to block  67 . The purpose of that block is to maintain the achieved power level of the ramp signal exactly at the desired nominal level A during the modulation period. Block  67  can be formed by a multiplexer (MUX), the output signal of which is locked to the input signal during the rise and fall periods of the ramp. Hence, when the ramps start to rise, selection signal SEL holds the MUX in a state which allows a signal incoming from scaler  66  to appear at the output port of the MUX. After the rising ramp has reached nominal power level A, selection signal SEL changes the state. In response to the change, a signal fed back from the output port to the input port of the multiplexer is directed again to the output port, whereby signal OUT remains constant. Accordingly, when the falling ramp starts, selection signal SEL changes its state again, whereupon the falling ramp is directed to the output port of MUX a  67 . 
     Power control is realized by scaling the ramp curve within the oscillator. The amplitude of the sinusoidal is controlled by factor A. For example, the downlink dynamic power control in GSM 900/DCS 1800 uses 16 power levels with 2 dB separation. The power control range is 0 . . . −32 dB, where 0 dB level is the nominal maximum power. The additional 2 dB range is reserved for gain stabilization of the transmitter analog parts. Furthermore, power control fine tuning step (0.25 dB) is introduced for this purpose. Therefore, the range of the initial amplitude value A* is from 0.0251 to 0.999. 
     If the ramp time is variable, then a fully parallel multiplier is needed. For applications with fixed ramp time, a fully parallel multiplier is not required, and it would indeed be a waste of silicon area. Multiplication by a fixed binary number can be accomplished with N−1 adders, where N is the number of nonzero bits in the coefficient. 
     If the clock frequency is 52 MHz, the. output frequency of the oscillator is 35.71 kHz and b is 25, then the coefficient 2 cos(2πf 0 /f clk ) is 1.99998137757162 (011111111111111110110001111) 2 . This requires 22 adders. 
     FIG. 7 shows the block diagram of a modified ramp generator and output power controller. In comparison with FIG. 6, it includes an extra two-input adder  78  and block  73 , which multiplies variable x1(n) with coefficient  2 . In blocks  74  and  75  the multiplication coefficients are 2(1−cos θ) and 1, respectively. In order to reduce the hardware complexity of the direct-form digital oscillator, we can write: 
     
       
         2 cos(θ)=2−2 −b1 └2 b1 (2−2 cos(θ))┘,  (23) 
       
     
     where                b1   =     [       log   2          1     2        (     1   -     cos                 θ       )           ]       ,           (   23   )                                
     and [r] is the smallest integer greater than or equal to r. The coefficient (2−2 cos(2πf 0 /f clk )) is 0.00001862 (000000000000000001001110000)2. The total number of adders to implement the coefficient 2 cos( 2πf   0 /f clk ) is reduced from 22 to 4. The coefficient is formed by multiplying the small fraction (2−2 cos(2πf 0 /f clk ) by the factor  2   b1 , where b1 is 16. This reduces hardware complexity by reducing the maximum word length needed in adders. The output of the adders must be multiplied by 2 −b1 , to keep the overall gain unchanged. The number of adders could be reduced further using the Canonic Signed Digit (CSD) numbers. 
     The error at the invented ramp generator output consists of two components 
     
       
           e ( n ) =e   1 ( n )+ e   2 ( n ),  (24) 
       
     
     where e 1 (n) is the error due to the ramp generator output quantization, e 2 (n) is the error that has been accumulated as a result of the recursive computations in the digital oscillator. 
     The bounds for e 1 (n) are given by 
     
       
         −2 −c   &lt;e   1 ≦0,  (25) 
       
     
     for truncation and 
     
       
         −2 −c /2 &lt;e   1 ≦2 −c /2,  (26) 
       
     
     for rounding and c is the fractional bits in the ramp generator and power level controller. 
     In the digital oscillator, besides the zero-input response y(n) of the second-order system, we get a zero-state response y err (n) due to to the random sequence e 2 (n) acting as an input signal. From equation (7) is obtained 
     
       
           y ( n+ 2) =αy ( n+ 1)− y ( n )+ e   2 ( n+ 2),  (27) 
       
     
     and by z transformation 
     
       
           Y ( z )= Y   ideal ( z )+ Y   err ( z ),  (28) 
       
     
     with Y ideal (z) due to (9). The z transform of the output error y err (n) is given by                    Y   err          (   z   )       =           z   2            E   2          (   z   )         -       z   2            e   2          (   0   )         -     z                     e   2          (   1   )               z   2     -     2      cos                   θ   0        z     +   1         ,           (   29   )                                
     with E 2 (z) being the z transform of the quantization error signal e 2 (n). Transforming Y err (z) back into the time domain results in an output error sequence                    y   err          (   n   )       =       1     sin                   θ   0                ∑     k   =   2     n                         e   2          (   k   )            sin        (       θ   0          (     n   -   k   +   1     )       )               ,                  for                 n     ≥   2     ,           (   30   )                                
     when e 2 (0) and e 2 (1) are assumed to be zero. 
     Equation (30) shows that output error is inversely proportional to sin θ 0 . Thus output error increases with decreasing digital oscillator frequency. If truncation is used, the right-hand side of equation (30) is negative since e 2 (k) is negative, (see equation (25)) and sin(θ 0 (n−k+1)) is positive, because the digital oscillator generates only half of the sine wave period. Therefore truncation results in very high output errors as shown in FIG.  8 A. 
     The fact that error is a deterministic signal forces us to investigate the worst case, which corresponds to the case where every truncation suffers from the maximum absolute error value. Thus the upper limit for the error becomes                          y     m                 a                 x                 err            (   M   )       =                    e     m                 a                 x         sin                   θ   0                ∑     k   =   2     M                     sin        (       θ   0          (     M   -   k   +   1     )       )                         ≈                  -     2     -   b           sin                   θ   0          sin        (       θ   0     /   2     )           ≈       -     2       -   b     +   1           θ   0   2         ,           ,           (   31   )                                
     where e max =−2 −b  is the worst case truncation error, b is fractional bits in the digital oscillator, 0&lt;θ 0 &lt;&lt;1, M=[π/θ 0 ], and [r] is the smallest integer greater than or equal to r. 
     FIG. 8B shows the error if rounding is used. The e 2 (k) gets positive and negative values, so the output error sequence gets lower values than in the case of truncation. Simulations indicate the accumulated error is below output quantization error when rounding is used, b is 25, and c is 12. 
     FIG.  9  and FIG. 10 show ramp up and ramp down profiles for transmitted time slots. Dashed lines show the time mask for the burst by burst power ramping. The curves fully satisfy the GSM 900/DCS 1800 masks. The power measured due to switching transients, which determines allowed spurious responses originated from the power ramping before and after the bursts, shall not exceed the limits shown in table I. The exact limits are given in GSM specification 05.05. The simulated power levels are well below the limits as shown in Table I. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 Spectrum due to switching transients 
               
             
          
           
               
                   
                   
                   
                 Simulated 
                   
               
               
                   
                   
                   
                 Maximum Power 
                 Simulated 
               
               
                   
                 Maximum power 
                   
                 (dBc) with 
                 Maximum Power 
               
               
                   
                 limit (dBc) 
                   
                 Raised 
                 (dBc) with 
               
             
          
           
               
                 Offset 
                 GSM 
                 DCS 
                 Cosine/Sine 
                 Blackman 
               
               
                 (kHz) 
                 900 
                 1800/1900 
                 Window 
                 Window 
               
               
                   
               
               
                  400 
                 −60 
                 −53 
                 −71.20 
                 −71.28 
               
               
                  600 
                 −70 
                 −61 
                 −78.09 
                 −79.69 
               
               
                 1200 
                 −77 
                 −69 
                 −84.97 
                 −86.35 
               
               
                 1800 
                 −77 
                 −69 
                 −86.23 
                 −89.13 
               
               
                   
               
             
          
         
       
     
     The oscillator can be implemented in various ways. The fixed coefficient multiplier in the sinusoidal oscillator in FIG. 5 could be replaced by a fully parallel multiplier, allowing the output frequency of the sinusoidal oscillator changed and the ramp duration time to be variable. 
     The ramp generator and power level controller according to the invention can also support a Blackman window. 
     FIG. 11 shows a ramp generator adapted to Blackman window. In that window equations (5) and (6) presented previously are of the form:                0.42      A     +     0.5                 A                   cos        (         π   *   t     )       T   r       )         +     0.08                 A                   cos        (       2      π   *   t       T   r       )                 (   32   )                                
     for the Blackman falling ramp, and                0.42                 A     +     0.5                 A                   cos        (           π   *   t     )       T   r       +   π     )         +     0.08                 A                   cos        (       2      π   *   t       T   r       )                 (   33   )                                
     for the Blackman rising ramp. 
     The frequency response of the burst signal with Blackman window is                W        (   f   )       =     V          ∑     n   =     -   ∞       ∞                       M        (     f   -     f   C     -     nf   g       )            [                 (     τ   -     T   r       )        sin                   c        (     n                     f   g          (     τ   -     T   r       )         )            cos        (     π                   T   r        n                   f   g       )           1   -       (     2        T   r        n                   f   g       )     2         -                 4        T   r        sin                   c        (       nf   g          T   r       )            cos        (     π                 n                     f   g          (     τ   -     T   r       )         )           25        (     1   -       (       T   r        n                   f   g       )     2       )               ]                   (   34   )                                
     where V is a proportional constant. This equation gives more attenuation of switching transients than raised cosine/sine switching (2). 
     The extra cosine term in equations (32) (33) requires one more digital oscillator in the ramp generator and power level controller as shown in FIG.  11 . 
     Recursive parts (see FIGS. 7 and 11) of the ramp generator and output power level controller determine the maximum sample rate. The clock frequency of the IF modulator is high so the multiplier is required to be fast. The idea of the implemented poly-phase ramp generator structure is to generate the desired sinusoidal oscillating signal with two oscillators, one of which spawns the odd samples and the other the even samples. This way both the odd and even oscillators can be operated at half the clock frequency. This means that four oscillators are needed to generate Blackman ramps. 
     FIG. 12 depicts a structure for generating Blackman ramps with four oscillators. The odd and even oscillator outputs are alternately selected with a 2-to-1 MUX, the select signal of which is the divided clock. The initial values of the poly-phase ramp generator are calculated by choosing first the same initial values as in the normal case and calculating the next two values using the difference equation (7) and choosing the odd samples for the odd oscillators and the even samples for the even oscillators.