INTRODUCTION
Delta sigma modulation with cascaded stages have been proposed (Gothenberg
and Tenhunen, 2002; AlTaweel, 2006). This approach
uses digital correction. Sensitivity of OTAs adaptation circuits used in high
precession analogue design would be reduced by careful design of the modulator
architecture.
Achieving high resolution and large bandwidth is accomplished using highorder
adaptive sigma delta modulators. The slopeoverload distortion for any varying
input signal, the higherorder noise shaping function was realized using cascaded
second order SDM. These circuits are sensitive to analog circuit imperfections,
a good matching between the digital output and the analog filter. A prominent
problem of the design of cascaded delta sigma modulation is the leakage of the
quantization noise (AlTaweel, 2006; Kiss,
1999). At last, the noise leakage can be reduced in the analog or digital
estimation circuit design (Jose, 2010; Zouari
et al., 2007).
Sigma delta quantizes estimate differently the quantization error of the first
stage which is quantized by multi bit second stage. Also this modulator requires
mismatchshaping DAC in the first stage. Cancelation of the imperfect error
causes the leakage of the quantization noise. To decrease this leakage, delta
sigma modulation with second order cascaded stages are used. In the first stage
of this scheme, 11.5 bit is applied whereas, 812 bit is applied in the second
stage. Also, an adaptive digital FIR filter is applied in this scheme (Hawksford,
2005; Rombouts et al., 2003; AlTaweel,
2010; Cauwenberghs and Temes, 2000).
This proposal consists of second order ASDM as shown in Fig.
1. The forthorder (22) cascaded modulator. Adaptive quantization with
backward estimation is used in all sections. This can be obtained by feedback
the encoded output signal adaptatively. As a result, the quantized input signal
amplitude is estimated continuously, with low OSR of 16, as shown in Fig.
2. The output pulse of encoded signal feedback through adaptation scheme
which consists of low pass filter followed by rectifier (OTAs) with variable
voltage gain to continuously estimating the amplitude of sampling at every moment
(Time varying the step size) to eliminate the varying input signal. If the input
signal X(n) is rapidly increasing or decreasing, the output encoded pulses Y(n)
are all 1s or all Os. Therefore the magnitude of the filter output will be relatively
large. This will control the amount of the amplifier bias current which will
vary the trans conductance (g_{m}), then the voltage gain of the amplifier
(A_{v}) which means time varying step size Δ(n), (AlTaweel,
2006).
In this study, a cascaded second order ADSM system is presented. In this approach, utilizing, adaptive quantization estimation and adaptive prediction estimation are used. High Operational Transconductance Amplifier (OTA) included higher gain, enhanced performance and better sensitivity for the cascaded structure than normal ADSM is proposed. Our method achieves continuous time varying step size and high SQNR with oversampling ratios. Simulation results show the illustration of the effect of linearity error quantization path for the cascaded structures.

Fig. 1: 
Second order ASDM section 

Fig. 2: 
Proposed cascaded two stages second order ASDMs scheme 
OVERSAMPLING ADCS
The ADC is suitable for high resolution lowspeed application, such as audio signal (0,33.4) KHz where the conversion speed is less than 0,3 KHz. For higher conversion speed 100150 MHZ, the sigma delta ADC is suitable and the resolution is approximately 1012 bit. So, the Nyquist rates ADCs are fast and flash ADC.
Oversampling improves the resolution obtained from a general Nyquist rate converter by using higher sampling frequency (fs>>2f_{m}), where oversampling ratio is defined as the ratio between the sampling frequency and the Nyquist frequency. So, if the samples occur often enough, the original waveform can be completely recovered from the sample sequence without fold over distortion by a lowpass filter. therefore the quantization noise power of an ADC can be reduced by factor of OSR, (OSR = fs/2f_{m}>>1) and for the maximum SNR there is a tradeoff between accuracy (N) and bandwidth (2f_{B}), Typically oversampling ratio ranges between 8 and 128.
SIMULATED QUANTIZATION NOISE
The quantization noise is calculated as a function of signal amplitude for
a coder with uniform quantization intervals. The quantization error is created
by digitizing an analog signal and is expressed as an average signal power to
average noise power, so, the quantization error sequence q(n), is limited to
±Δ/2 and the quantization interval implying a uniform probability
density of amplitude 1/Δ (Limketkai and Victor, 2000;
Ziemer and Tranter, 1995). The digital quantizer output
y(n) is fed back through DAC which is converted back into an analog discrete
time signal X(n). So, the quantization error sequence q (n) is:
where, e_{qi} and e_{qo} are the input and output quantizer, the a mplitude of the quantization error for the resolution N = 4 bits:
where, A is the peak amplitude of the input sin wave signals. According to the entire signaltonoise quantization the ratio is given as:
Substituting the total number of intervals into Eq. 4 produce the performance equation SNR as follows:
where, A_{max} is the maximum non overloaded amplitude, it's noted
from Eq. 5 there is a relationship between the resolution
N and its SNR performance in dB (Kiss, 1999; Jose,
2010).
In this study, cascaded two stages second order is introduced. Each stage converts the quantization error of the previous stage, then using the output of each stage to construct the output signal with little quantization errors of all stages as shown in Fig. 3.
The noise shaping can be achieved by recombination, the output of each stage
of Fig. 2.

Fig. 3: 
Comparative SNR performance of the2O ASDM and cascaded 2O
ASDM 

Fig. 4: 
SNDR as a function of input amplitude 
Singlebit ADC is employed in the first second order stage and four bits ADC
is employed in the lower resolution to improve the dynamic range, where the
dynamic range is the ratio of the maximum amplitude signal to minimum amplitude
signal, therefore, the ideal dynamic range is 30 dB. SNDR as a function of input
amplitude is shown in Fig. 4.
Simulation shows that the maximum signaltoquantization noise ratio for the
cascaded structure achieved with higher performance and greater dynamic range
(Ziemer and Tranter, 1995; Bellamy,
2000):

Fig. 5: 
SNR vs. oversampling ratio for fourth order cascaded ASDM 
Using Eq. 5 or 6 the simulated SNR versus
the oversampling ratio for fourth order cascaded shown in Fig.
5.
CONCLUSION
In this study, a cascaded second order ASDM is introduced. The noise leakage is cancelled by using adaptive analog circuits. The modulator employs 1 bit level quantization and 4 bits quantization in the final stage of 22 cascaded circuit, also the SQNR performance is achieved at low OSR, in order to increase the SNR and SNDR at oversampling ratio using adaptive circuit for each cascaded stage.
The simulation results for the proposed architecture were highlighted using a good OTA parameters with both large bandwidth and open loop gain. The proposed system achieved better SNR of 95 dB at X[n] = 4 dB and SNDR of 89 dB at X[n] = 4 dB. In addition to that an acceptable SNR was obtained for low OSR with good performance (N = 8).