Essay Sample on Cascading Methods

Introduction

Designing of the composite filters which have transition bands can be done using cascade connection. The use of the cascading is essential for the creation and execution of complex data for the coding and processing of the workflows (Shiung, Yang & Yang, 2016). Therefore, cascading requires algorithm use. According to Lu & Hinamoto (2017), an algorithm is used to optimize the prototype and shape filters in different steps coupled with sequential designs. A composite filter is a digital system for filtering which consists of subfilters which have cascade. Shaping filters are constructed using cascading tricks. In a study, Shiung, Ferng & Lyons (2005) asserted that a comb filter is essential for the removal of the dc offset and eliminate the complexities in filtering. The advantage of the comb filter is that it is simple in implementing and can reduce the problems in dc off-setting. Therefore, this report endeavored to determine cascading tricks useful for designing of the composite filters which have narrow and broad bands.

Is your time best spent reading someone else’s essay? Get a 100% original essay FROM A CERTIFIED WRITER!

Save 25%

Digital Filters

Whitehouse and Reason (1965) digitally simulated the 2RC filter to comprehend the phase behavior and to resolve the issue. They used a weighing function that relied on the cutoff to describe the filter. According to Gupta (2012), a cutoff plays the same role as the size of a component in the graphical examination. It separates the short wavelength sections from the long-wavelength ones (Wang, Shi, & Peng, 2016). Whitehouse and Reason were able to demonstrate that the raw profile could be combined with the weighting function to develop a running average mean line. This way, the acquired digital mean line was similar to the hardware electrical filter mean line (Gupta, 2012).

Undoubtedly, the digital execution of the 2RC filter has led to critical progress. Given the rising abilities of computers, the electrical filter rapidly became outdated, and filtering began to be practiced by the use of computers. Raja, Radhakrishnan, and Muralikrishnan (1977, 1979, 2002) have widely studied the digital filtering method. They have previously applied the Fourier transform for filtering surface profiles in their work. Next, to that, they also proved to filter in the time domain utilizing impulse reaction. The advancement of digital filtering gave room for researchers and scientists to look for resolutions to the main disadvantages of the 2RC filter, that affected its non-linear phase. As per Gupta (2012), early researchers designed the phase-correct 2RC filter.

Digital filters have a critical job in the design of composite filters with sharp transition bands. Next, to that, they are presently being favored over their older partners in numerous application areas because of its diverse merits that comprise of: increasingly linearity phase reaction, more straightforward proliferation, repeatable execution, increased robustness to noise and ecological changes, programmability, and adaptability (Gupta, 2012), significantly wider operating frequency range, and provision for saving data for future application (Moysis et al., 2015).

According to Nongpiur, Shpak, and Antoniou (2013), two general types of digital filters include the Finite Impulse Response (FIR) filters and the Infinite Impulse Response (IIR) filters. Below are the corresponding equations that express the two types of filters:

yn=k=0hk.xn-k...IIR Filter... Eq. 1

yn=k=0N-1hk.xn-k...(FIR Filter)... Eq. 2

where x(n) represents the input sequence, y(n) represents the filtered output sequence, and h(n) represents the system's impulse reaction (Lu & Hinamoto, 2016).

It is important to note that it is impossible to implement the IIR filter equation in a PC directly. Instead, we implement its recursive form which is referred to as the Difference Equation (Wang, She & Peng, 2016) for the IIR filter. The equation is expressed below:

yn=k=0Nakxn-k-k=1Mbky(n-k)

Where ak's and bk's represent the numerator and denominator polynomial coefficients of the rational Z-transfer function respectively (Gupta, 2012). Otherwise, the IIR and FIR filters can be represented by their Z-transfer functions in the forms expressed below:

Hz=k=0Nak.z-11+k=1Mbk.z-k...IIR Filter... Eq. 3

Hz=k=0Nhn.z-k...FIR Filter... Eq. 4

IIR Filter Design using the Pole-Zero Placement Approach

According to Gupta (2012), the pole-zero placement approach is an easy and simple method of designing an infinite impulse response (IIR) filter deliberately putting poles and zeros on the z-plane, mulling over that the recurrence reaction of the framework can be seen along the unit circle shape. In principle, a zero put along the unit hover at an edge compared to a specific recurrence would result in a reaction of zero at that recurrence. A pole anyway set close to the unit circle would create a large point or addition at the recurrence compared to the edge at where the pole is put.

Modern Improvements in Filtering

In the most recent decade, there have been some noteworthy advances in sifting systems; audits can be found in Raja et al. (2002) and Muralikrishnan (2003). The Gaussian filter, which is broadly utilized today, experiences a few deficiencies. Initially, as per Matthe et al. (2015), the waviness profile has bending close to the edges (this is valid for open profiles, not for roundness).

Properties of Cascade Filters

Overall, IIR filters have much sharper transition bands compared to FIR filters of similar nature. This paper focuses on the design of composite filters that have sharp transition bands by cascading FIR and IIR filters as shown in figure 1 below.

Figure 1: Combining two figures in a cascaded form. Source: Shiung, Yang, and Yang (2016).

The frequency reaction of the prototype filter is enhanced by utilizing a shaping filter. The shaping filter can either be a comb or a complementary comb filter. According to Shiung, Yang and Yang (2016), the resulting transfer function of the two cascaded filters is the product of those functions expressed in the equation below.

Hcas (z) = Hsha (z) Hpro (z)... Eq. 5

Where Hsha (z) and Hpro (z) represent the transfer functions of the shaping and prototype filters respectively. Xz and Y(z) represents the system's input and output transfer sequences respectively. Given its instability, IIR filters are normally found by applying an order of only two (Shiung, Yang & Yang, 2016).

Nonetheless, as Shiung, Yang and Yang (2016) argue, different kinds of IIR filters that happen to have an overshoot at the passband edge exist. These are referred to as the Gibbs phenomenon. This overshoot is usually viewed as a disadvantage and is difficult to dodge in the design of filters. One of the possible tricks to use is the application of comb or complementary comb filters for remunerating the overshoot and to create a composite filter with a level passband reaction. Given that both the shaping and prototype filters are linear, the recurrence reaction of the cascade filter is unaltered ihatf we reorder the cascade of filters. If any phase of shaping filter is insufficient to meet the structure details, we can cascade the prototype and two shaping filters of proper lengths.

Properties of Shaping Filters

The transfer function of shaping filters can either be expressed as:

Hsha (z) = H1 (z) 1 - z-k... Eq. 6 or

Hsha (z) = H2 (z) 1 + z-k... Eq. 7 where k = 1, 2, 3, ..., n = 9.

According to Matthe et al. (2014), both picks of Hsha (z) are either symmetrical or asymmetrical with a linear phase response. In equations [2] and [3] above, H1 (z) denotes the comb filter whereas H2 (z) denotes the complementary comb filter. The frequency responses of both filters can be easily computed by substituting z = ejw into the respective equations. It is important to note that the number of nulls increases concurrently as K increases. 6 Step 6Here, in the final step, we express the time domain equations for both the shaping and prototype filters as;

yn=b0xn+b1xn-1+...+bNxn-N+a1yn-1+a2yn-2+...+aMy(n-M)...Eq. 8 and

yn=x(n)x(n-K)...Eq. 9 respectively where;

xn and y(n) denote the input and output sequence of each filter. It is important to note that both filters are linear and time-invariant hence making it possible to change their filtering sequence.

The cascading methods expressed in the above steps lies in initially identifying the filter orders and then fitting the specified design definitions by trimming the prototype filter frequency response. This is an easy and effective method for solving | Hcas (op) | = | Hsha1 (op) | | Hsha2 (op) | | Hpro (op) | = | Hpro (0) |. Stages 1 to 3 above attempt to locate an appropriate shaping filter through an underlying supposition for the prototype filter. After setting the two shaping filter orders, we continue to trim the prototype filter, so the design determinations are met. Cascading more shaping filters enhances the possibility of achieving sharper transition bands.