An essential component of modulators is the Fast Fourier Transform. Many OFDM modulations and demodulation systems have been developed utilizing floating-point FFT processors for fast convolution and spectrum estimation. Floating-point arithmetic has been widely implemented in DSP applications, making the process highly attractive to the designer and end-user. Algorithms and structures that do not have to deal with algebraic difficulties have gotten people's attention. In DSP, the Fast Fourier Transform (FFT) approach can be used to execute DFT calculations rapidly and reliably. Nikhilam Sutra and Urdhva tiryagbhyam, the binary number system, was originally utilized with the sutras and is now used for the construction of many digital gadgets, and these two approaches are discussed in this article. The Nikhilam navatascaramam Sutra explains how to subtract a number from the 10thspeed up to a thousand. For it to be appropriate, at least one operand must be close to the power of 10. Because it is compatible with the binary number system, urdhva-tiryakbhyam (Vertical and across) is a widely accepted approach.

**Vedic mathematics**

According to its origins in the Vedas, "Vedic Mathematics" is known as such. It's a time-honored method for making algebraic and arithmetic calculations easier. Sri Bharati Krishna Tirthaji Maharaj resurrected it from the Vedas.

Table 1 reveals that Tiryagbyam is horizontal whereas Urdhva is vertical. The two-bit multiplication includes two binary numbers (A1A0 and B1B0) known as multiplier and multiplicand (S0 and S1). This is done by multiplying B1 and A0, next multiplying A1 and B0, then multiplying C0, and finally concatenating all the bits together to yield, S2, and S1.

Algorithm 1 shows that the restructuring of the traditional multiplication approach is used in the Vedic algorithm. The goal of this project is to provide users with the basic foundation of Vedic formulas, which may be a very important asset for them in terms of clear comprehension. Starting with 2-digit multiplicand and 2-digit multiplier multiplication, the Vedic Sutra (Algorithm) is applied. In addition, alphabetical symbols must be used to represent numerals. Later on, this will be expanded to discover products of a multi-digit multiplicand by another multi-digit multiplier.

**Fast Fourier Transform (FFT)**

The FFT is a classifier for every method that evaluates the DFT with the computational complexity of This method uses a divide-and-conquer strategy to split the DFT calculation over two more manageable issues. Most widely used repeated FFTs make use of a rearranging method where every other component is switched out for its reversing binary value. This makes it possible to execute in-place calculations, in which the DFT results are immediately substituted for the components of the original vectors. Based on the way it is used, the FFT may take the form of decimation in time (DIT) or decimation in frequency (DIF), which is a derivative of the DFT's traditional use in digital signal processing (DSP) application.** **

**a) 8-bit FFT architecture **

In this instance, the samples of the data input seem to be out of sequence, while the samples of the result are in sequence. In figure 3, the various jiggle components are mostly indicated. Below is a simple signal flow diagram for an 8-bit FFT.Figure 2 demonstrate the signal flow graph of 8-bit FFT architecture.

**b) 16-bit FFT architecture**

There are many jiggle factor configurations employed in each butterfly phase. The jiggle factor was pre-calculated independently and kept in a variable of type "complex array" since it is a constant term. A jiggle factor is a fractional value that is expressed in 16-bit signed integers; bit 15 denotes the sign, bit 14 denotes the integer portion, and bits 13 to 0 denote the fractional portion. It is derived using the following equation:

The butterfly inputs are multiplied by the jiggle factor, and because of how the "mult" tool was built; the output of the multiplication simply takes the integer portion into account. In doing so, we continued using integers while gaining precision from the jiggle factor, which also improves the outcome. Figure 3 displays a schematic of the 16-point FFT method along with the WN utilized for each step of the butterflies.

**Sutras**

Multiplying is not required in each signal processing application.According to this research, multipliers demand more silicon space and also waste more power as a result of the switching activity; therefore they generate the longest route through adders and transports. The processor's top range is therefore limited. Vedic Multipliers use distinct sutras for diverse multiply algorithms to bypass the limitations of critical path and power in CPUs. When employing alternative Sutras, however, the Vedic Multiplier can minimize both critical path and dynamic power. Mentioned following are the two sutras utilized for Vedic multiplication:

- Urdhva Tiryagbhyam
- Nikhilam Navatascaramam

**i) Urdhva Tiryagbhyam sutra**

The Urdhva Tiryagbhyam multiplier is a Vedic mathematics technique developed in India during the Vedic period. It's a formula and mostly "vertical and crosswise" multiplier-related if we talk about the sutras. The half consequence of continuous addition was formed and completed in this case. Many multiplication problems may be solved using Urdhva Tiryakbhyam. It can also be used to divide a big number by a smaller number, but it is beyond the scope of this paper. For centuries, the decimal number system has been multiplied using this Sutra. All partial products may be generated simultaneously with the inclusion of these partial products, thanks to a revolutionary approach. To make the technique broader, it may be applied to n * n bits.The urdhava tiryagbhyam technique is described in Algorithm 2 and may be applied with variables such as x, m, z, t, etc. We must first multiply the provided matrix, and then record the remainder as carry. Multiply the integers in the 2*2 matrix and add the carry. Repeat the method until you reach the desired result.

The binary digits method is used with the Urdhva Tiryakbhyam Sutra to produce a digital multiplier design. The frequently utilized array multiplier technique is comparable to this. For two n-bit values A and B, where A = a1a0 and B = b1b0, this approach is explained. We define multiplicands and multipliers. Then add the LSB of the output achieved to the LSB of each of them. After that, multiply the LSB of the multiplier by the second bit of the multiplicand. The next step is to multiply the LSB of the multiplicand by the multiplier's second bit. The second piece of the outcome is obtained by adding them. Then divide the LSB of the multiplicand by the third bit of the multiplier. The third bit of the multiplicand is multiplied by the LSB of the multiplier, followed by the second bit of the multiplicand and the second bit of the multiplier. To get the third bit of the outcome, just add this. Once again divide the LSB of the multiplier by the second bit of the multiplicand. Then divide the LSB of the multiplicand by the multiplier's second bit. Then include them. The third piece of the result was obtained. The last step is to multiply the LSB of the multiplicand and multipliers to get the outcome.

Figure 4 shows how to multiply 4-bit by using the sutra. Z [3 to 0] is the multiplier (lower values), while V [3 to 0] is the multiplicand (upper values). At each step, the sum is a partial product with no zeroes added. After zeroes have been inserted, the incomplete product is called temp. To achieve the final result t, all of the partial products are merged.

**ii) Nikhilam navatascaramam **

The NikhilamNavatascaramam Sutra is generally referred to as "all from 9 and the end from 10". This method may be used with any number that is closer to a base value than a power of 10, such as 10, 100, 1000, and so on. Using this strategy, we may multiply from both a theoretical and a practical foundation. The base value may be any one of the 10s, and the working base can be a multiple of 10, 100, 1000, etc. Depending on the base value, the input numbers might be more or less when multiplied. Large numbers may be efficiently multiplied using this approach. The base numbers are multiplied by the complements of the numbers contained here. The sum of two big numbers is thus reduced to the sum of their complements. Verify that the resulting full values are less than the actual values. Figure 5 shows the flow chart for nikhilam navatascaramam sutra.

For subtracting the provided integer, this method employs the closest power of the base value. Multiply the acquired values once more and add the result value to the total. It is a straightforward approach for multiplying data that also saves time.Two-digit multiplication may be accomplished in only one multiplication, whereas the traditional method requires four multiplications. In Figure 6, the Nikhilam technique for multiplying two-digit values whose base is smaller than the closest base is shown. Now let us take an example of multiplying 98 by 99.

The following are the stages involved in multiplying two numbers.

- To begin, remove the base's nearest power from the multiplier's and multiplicand's product.
- To get lower integers for both operands after subtraction, choose the base power that is the closest to it.
- It is then multiplied by two to arrive at the final product's two least significant digits.
- To get the final product's most significant two digits, subtract the base's nearest power from the multiplier or multiplicand.