In communications field, sampling theorem is a condition described by Harry Nyquistand Claude Shannon [2, 3], to be able to convert a band limited signal (continuous-time) into a digital sequence (discrete in time) samples. This method is also called Nyquist- Shannon theorem. The theorem can be simply understood as: The sampling frequency should be at least twice the maximum bandwidth of the signal, to not lose information contained in the signal. Mathematically, Fs ≥ 2B (1.1).
with Fs the sampling frequency, B the bandwidth of the signal, can be considered as the highest frequency which is contained in the signal, and FNyq = 2B is also called Nyquist rate. Back to the context of spectrum sensing, in case the sensed signal raises up to several tens GHz, it will lead to the problem that the sampling rate overcomes the capability of a conventional sampling hardware. As explained above, a sampling hardware is able to reach up to several GHz sampling, however, it requires an extra power consumption, a high memory to store the output sample, and certainly a high cost to deploy. In real life applications, it is impossible to implement such a system for a huge quantity of devices necessary for the largest number of users. Consequently, methods need to be proposed to reduce the sampling rate, at sub-Nyquist rate without loss of information.
Papoulis generalized sampling
A very first method w hich was p roposed i n 1 977 b y P apoulis , i s a generalized multichannel scheme to sample a bandlimited signal at sub-Nyquist rate. Considering that the Nyquist rate of wideband input is equal M times the ADC sampling rate Fs. In this scheme, M converters are deployed in parallel. At each channel, the bandlimited signal is sampled at a sub-Nyquist rate Fs. Thus, the sum of all channels sampling rate is equal to the Nyquist rate. This method is also called uniform interleaved sampling method.
The uniform interleaved sampling scheme provides a solution to deal with the exceeding sampling rate of a converter. Although this model is not considered as a Compressed Sensing scheme, it provides a basic overview of Compressed Sensing methods. To implement a uniform interleaved sampling scheme, a multiple-delay method is applied as in To fully collect all samples of signal x(t), the output of the second channel must come later than the output of the first channel, a Ts/M time space and so on for the other channels. After that, all samples are multiplexed to have a digital output of signal x(t). By taking care of this delay, this system sampling rate operates as if it is multiplied by M times, that is to say MFs.
Compressed Sampling of sparse signal
A signal x(t) is called sparse in some representation bases or dedicated bases, if and only if there is a small number of non-zero components in the well chosen representation base. Assuming that signal x(t) has d-dimension and s number of non-zero elements, whereas s ≪ d. Let us denote P = [p1; p2; · · · ; pN] the measurement matrix, where vector pi (i ∈ [1,N]) has the same size with input vector x and the size of measurement matrix P is N × d. The measurement y is calculated by y = Px. (1.2)
The term of compression is explained as, the size of vector y is N × 1, it is much smaller than vector x (d × 1). That is to say, instead of sampling the signal x at a high rate and then compressing its sampled data, the data of x can be sensed directly in its compressed form y, and sampled at a lower rate. The last part is a brief introduction about compressed sampling reconstruction methods. Figure 1.5 shows an illustration of the CS equation with the size of x, P and y.
Sub-Nyquist schemes for cognitive radio
A CS acquisition system is a combination of techniques that acquires a wideband signal in an economical manner, while all the useful information is stored and processed. As in the previous section, sampling methods must satisfy the two constraints: ADC rate and ADC bandwidth. Consequently, this section introduces some methods which exist in practice and should be considered as the premises to come up with a new approach in the wideband monitoring system.
Non-uniform interleaved sampling: Multicoset sampling
Similar to uniform interleaved sampling, non-uniform interleaved sampling  is proposed to handle the problem of sampling rate. However, the delays in this model are not regular in comparison with the uniform interleaved sampling, as shown in Figure 1.6. As in the uniform interleaved sampling, by taking the delay very well, the sampling rate is considered to be reduced since it does not need to sample the entire analog signal. Nevertheless, this scheme also has the problem of input signal bandwidth. The advantage of this scheme is simpler than the uniform interleaved sampling, since the exact delay time to synchronize outputs is not necessary.
One of the methods which is developed based on non-uniform and interleaved sampling model is multicoset sampling [46, 48], this scheme is shown in Figure 1.7.
MWC: An equivalent model
With the size of matrix P and lengths of vector y and z that are found in Equation 2.15, the MWC system can be depicted by an equivalent scheme  as in Figure 2.3. In practice, the MWC has M physical channels. When M is a high number, it is difficult to deploy the MWC due to the high-cost and complex system, since each channel has one mixer, one filter and one ADC. It seems to be a constraint to go further in experimental study. Indeed, a prototype of MWC now could be expanded to 4 physical channels [6, 22].
In the theoretical point of view, however, the equivalent scheme has the same functions as the MWC scheme but the mathematical expressions are different. Hence, the equivalent model can bring a better study on the MWC and its components. The equivalent scheme will provide exactly the same outputs as MWC because the same transfer functions are used. It is important to note that this scheme is a theoretical model, which cannot be applied in practice. The goal of this model is to deeply understand how the system works by seeking the details of every component such as how many channels M should have exactly. The sampling frequency Fs, sequences pi(t) and its repetition frequency Fp need to be examined.
Going further in details of each component, at first, supposing that pi(t) is an arbitrary waveform and it is not periodic. It leads to the infinite sequence of outputs after mixing that cannot be managed. To solve this problem and make it easier, the mixing function pi(t) is chosen as a periodic pseudo-random waveform, with the period Tp. Consequently, there is a finite output xi(t) at each physical channel. Suppose that pi(t) is also set as a piece-wise function with values varying between ±1. The mixing function pi(t) is illustrated in Figure 2.4. Moreover, the sampling period of the conventional ADC is Ts. In the Nyquist bandwidth FNyq, let us denote L the ratio between FNyq and Fp, with L = FNyq/Fp. It means that in the Nyquist bandwidth, there are L subbands with Fp bandwidth. The goal of theMWC is to detect the active subbands l where the transmitters of input x(t) are located, with L = 2L0 + 1 and −L0 ≤ l ≤ L0.
Number of active subbands estimation
As explained in the previous section, in the MWC, the Nyquist bandwidth is divided into L subbands. The active subbands are the subband which the transmitters are located in. In this section, some estimation methods for the number of active subbands are introduced. This task is considered as a pre-processing step in Figure 2.1, before spectrum reconstructing. At the output of the MWC, the vector yi[n] (1 ≤ i ≤ M) has N sub- Nyquist samples. Then, this output is considered as a M × N matrix y, or qM × N in the case of using collapsing factor q. Similarly, the signal z[n] has N sub-Nyquist sample elements and is considered as a L × N matrix. The main perspective of the MWC is to seek for z when the output y and the sensing matrix P are known.
The number of active subbands estimation methods are based on the singular values of matrix y ∈ CM×N , or equivalently the associated autocorrelation matrix eigenvalues. Indeed, their variation is directly related to the number of active subbands s. Moreover, there are exactly s non-zero eigenvalues in the noiseless case. In the case of additive whiteGaussian noise, all the M − s values which are smaller than the eigenvalues, are equal to the noise variance. In the case of very strong noise, it is difficult to determine exactly the number of active subbands.
The methods Akaike Information Criterion (AIC)  and Minimum Description Length (MDL)  are easy to adapt with the MWC system for estimating the number of active subbands. The noise is considered as additive white Gaussian noise (AWGN). The number of subbands is determined by the best match between the model and the observation elements of MWC output s = arg min k [C(k)], (2.34)
Table of contents :
List of Figures
List of Tables
1 Compressed Sensing/Compressed Sampling
1.1 Cognitive radio, Spectrum Sensing and Compressed Sensing concepts
1.1.1 Nyquist-Shannon theorem
1.1.2 Papoulis generalized sampling
1.1.3 Compressed Sensing
1.1.4 Compressed Sampling of sparse signal
1.1.5 Compressed Sensing applications
1.2 Sub-Nyquist schemes for cognitive radio
1.2.1 Non-uniform interleaved sampling: Multicoset sampling
1.2.2 Random Demodulation
1.3 Reconstruction algorithm CS
1.3.1 ℓ1-minimization algorithms
1.3.2 Greedy algorithms
2 MWC: From theory to practice
2.1 Theoretical background
2.1.1 MWC: physical scheme
2.1.2 MWC: An equivalent model
2.1.3 MWC parameter conditions
2.1.4 Number of active subbands estimation
2.1.5 Input signal reconstruction and performance evaluation
2.2 Transmitter signal key parameters on the CS spectrum reconstruction .
2.2.1 Minimum bandwidth of one transmitter
2.2.2 Number of transmitters
2.2.3 Frequency spacing and LoRaWAN spectrum detection application .
2.3 Hardware implementation in the literature
2.3.1 Analog boards
2.3.2 MWC components imperfections
2.3.3 Hardware Calibration
2.4 Lab-STICC Prototype and testbed
3 Implementation of non-ideal lowpass filter on the MWC
3.1 Non-ideal lowpass filter in the MWC
3.1.2 Analog filters
3.1.3 Post-processing method and Simulation results
3.1.4 Different parameters simulations
3.2 COTS lowpass filter with global compensation for the MWC
3.2.1 Implementation and characterization of SXLP-36+ real filter
3.2.2 Simulation results
3.3 Practical results with testbed
4 Low-Bit Quantization Methods for the MWC CS
4.2 Low-bit memoryless ADC and Oversampling method
4.2.1 Low-bit memoryless ADC
4.3 ΣΔ Analog-to-Digital Converter
4.4 Simulation Results
Conclusion and Future Works