# 电气工程代写|信号和系统代写signals and systems代考|EECS3210

## 电气工程代写|信号和系统代写signals and systems代考|Decimation in Time (DIT) Radix-2 FFT

Consider Eqs. (2.71) and (2.72) which are given below.
$$\begin{gathered} X(k)=\sum_{n=0}^{N-1} x(n) \mathrm{e}^{(-j 2 \pi k n) / N} \ x(n)=\frac{1}{N} \sum_{k=0}^{N-1} X(k) \mathrm{e}^{(j 2 \pi k n) / N} \end{gathered}$$
In direct evaluation of spectral components, the number of complex multiplications and additions required are $N^2$ and $N(N-1)$ respectively as stated in Sect. 2.6. Such a huge number of mathematical operations limit the $\mathrm{BW}$ of digital signal processors. Classical DFT approach does not use the two important properties of twiddle factor, namely symmetry and periodicity properties which are given below:
\begin{aligned} W_N^{k+N / 2} &=-W_N^k \ W_N^{k+N} &=W_N^k \end{aligned}
Radix-2 FFT algorithm exploits these two properties thereby removing redundant mathematical operations. However the results obtained using FFT algorithms are exactly the same as that of DFT. Further, the efficiency of FFT algorithm increases as $N$ is increased. For example, if $N=512$, DFT requires nearly 110 times more multiplications than FFT algorithm. The basic principle of FFT algorithm is therefore to decompose DFT into successively smaller DFTs. The manner in which this decomposition is done leads to different FFT algorithms. The two basic classes of algorithms are:

1. Decimation in time (DIT)
2. Decimation in frequency (DIF)

## 电气工程代写|信号和系统代写signals and systems代考|Use of the FFT Algorithm in Linear Filtering and Correlation

An important application of the FFT algorithm is in FIR linear filtering of long data sequences. The response of an LTI system for any arbitrary input is given by linear convolution of the input and the impulse response of the system. If one of the sequences (either the input sequence or impulse response sequence) is very much larger than the other, then it is very difficult to compute the linear convolution using DFT for the following reasons:

1. The entire sequence should be available before convolution can be carried out. This makes long delay in getting the output.
2. Large amounts of memory is required to store the sequences.
The above problems can be overcome by linear filtering of longer sequence into the size of smaller sequences. Then the linear convolution of each section of
3. longer sequences and the smaller sequence is performed. The output sequences obtained from the convolutions of the sections are combined to get the overall output sequences. There are two methods to perform the linear filtering. They are,
1. Overlap add method.
2. Overlap save method.

# 信号和系统代考

## 电气工程代写|信号和系统代写signals and systems代考|Decimation in Time (DIT) Radix-2 FFT

$$X(k)=\sum_{n=0}^{N-1} x(n) \mathrm{e}^{(-j 2 \pi k n) / N} x(n)=\frac{1}{N} \sum_{k=0}^{N-1} X(k) \mathrm{e}^{(j 2 \pi k n) / N}$$

$$W_N^{k+N / 2}=-W_N^k W_N^{k+N} \quad=W_N^k$$
Radix-2 FFT 算法利用了这两个特性，从而消除了冗余的数学运算。然而，使用 FFT 算法获得的结果与 DFT 完全相同。此外，FFT 算法的效率随着 $N$ 增加。例如，如果 $N=512$, DFT 需要比 FFT 算法多近 110 倍的 乘法运算。因此，FFT 算法的基本原理是将 DFT 分解为连续更小的 DFT。这种分解的方式导致了不同的 FFT 算法。两种基本的算法类别是:

1. 时间抽取 (DIT)
2. 频率抽取 (DIF)

## 电气工程代写|信号和系统代写signals and systems代考|Use of the FFT Algorithm in Linear Filtering and Correlation

FFT 算法的一个重要应用是长数据序列的 FIR 线性滤波。LTI 系统对任意输入的响应由输入的线性卷积和系统的脉冲响应给出。如果其中一个序列（输入序列或脉冲响应序列）比另一个大得多，那么使用 DFT 计算线性卷积非常困难，原因如下：

1. 在进行卷积之前，应该可以使用整个序列。这使得获得输出的延迟很长。
2. 存储序列需要大量内存。
上述问题可以通过将较长序列线性过滤成较小序列的大小来克服。然后对每一部分的线性卷积
3. 执行较长的序列，执行较小的序列。将各个部分的卷积得到的输出序列组合起来得到整体的输出序列。有两种方法可以执行线性滤波。他们是，
1. 重叠添加方法。
2. 重叠保存方法。

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