# Review of Channel Estimation for Candidate Waveforms of Next Generation Networks
## 2 Structure of Filter Bank Multi-Carrier Systems  
One of the most fundamental building blocks of filter bank multi-carrier waveforms is a well-designed prototype filter. Therefore, this section presents a review of the filters and the fundamental building block of FBMC based transceivers. A filter can be defined as a frequency selective circuit that is designed to pass signals of a particular frequency band while attenuating signals of other frequencies [22]. Filters are generally employed to eliminate or minimise unwanted system features such as interference, noise and distortions. Filters are classified in many ways, which could depend on the functions they are required to perform or be based on their physical structures. Hence, filters may be broadly classified as analogue/digital filters (based on function), active/passive filters (based on structure), etc. Passive filters are filters that are designed with the use of passive components such as resistors, capacitors, inductors, relays, etc. Active filters are designed with the aid of active components such as transistors, integrated circuits, diodes and passive components. Digital filters also known as discrete-time filters are employed in digital signal processing (DSP), mainly for the separation of digital signals that are combined together and for the restoration of distorted signals. The implementation of digital filters can be realised by convolution or by recursion. Convolution filters are known as finite impulse response (FIR) filters while recursion filters are known as infinite impulse response (IIR) filters. An FIR filter is a digital filter that has a finite duration of impulse response. These filters are reverse cases of the IIR filters since there is no feedback of output signals into the input. As such, they can be called non-recursive filters [22; 23]. The impulse responses of these filters are finite because of the absence of feedback. Some of the methods used for designing FIR filters include windowing, frequency sampling and equiripple approximation [22; 24]. The finite length impulse response of the FIR filters is obtained by truncating an infinite length impulse response. The truncation of an infinite duration sequence to yield a finite response is termed windowing. Some functions are used to perform windowing. Examples of such functions include rectangular, Barlett, Hanning, Hamming and Blackman windowing functions. Infinite impulse response filters are digital filters that have an infinite duration of impulse response. They are also called recursive filters because they are designed to use the feedback signals obtained from the system output. These types of filters have nonlinear phase and are usually unstable. Some examples of these filters are Butterworth filters, Chebyshev type I and Chebyshev type II filters, Elliptic filters and Bessel filters. Other examples of this kind of filters are described in [22; 23]. All filters are classified based on their frequency response characteristics [22; 25]. In the design of digital filters, the frequencies that are allowed to pass through are known as pass-band frequencies [25], whereas the rejected frequencies can be referred to as the stop-band frequencies. Based on this information, filters are classified as low-pass filters, high-pass filters, band-pass filters, band-stop filters (band-reject filters) and the all-pass filters. Low-pass filters allow signals of low frequencies to pass through while rejecting signals above its cut-off frequency. High-pass filters eliminate frequencies below its cut-off frequency while allowing signals of high frequencies to pass through. Band-pass filters allow signals within a certain range of frequencies to pass through while attenuating all other frequencies. Band-stop filters allow all frequencies to pass through while attenuating a certain range of frequencies. Finally, the all-pass filters allow all frequencies to pass through; however, they change the phase of the passing signals without changing their amplitudes.