FFT算法的C语言实现

：数字信号处理

需要注意的几个点

#mermaid-svg-Q0Cv61uzu3GVxhM0 .label{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);fill:#333;color:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .label text{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .node rect,#mermaid-svg-Q0Cv61uzu3GVxhM0 .node circle,#mermaid-svg-Q0Cv61uzu3GVxhM0 .node ellipse,#mermaid-svg-Q0Cv61uzu3GVxhM0 .node polygon,#mermaid-svg-Q0Cv61uzu3GVxhM0 .node path{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .node .label{text-align:center;fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .node.clickable{cursor:pointer}#mermaid-svg-Q0Cv61uzu3GVxhM0 .arrowheadPath{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edgePath .path{stroke:#333;stroke-width:1.5px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .flowchart-link{stroke:#333;fill:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edgeLabel{background-color:#e8e8e8;text-align:center}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edgeLabel rect{opacity:0.9}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edgeLabel span{color:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .cluster rect{fill:#ffffde;stroke:#aa3;stroke-width:1px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .cluster text{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:12px;background:#ffffde;border:1px solid #aa3;border-radius:2px;pointer-events:none;z-index:100}#mermaid-svg-Q0Cv61uzu3GVxhM0 .actor{stroke:#ccf;fill:#ECECFF}#mermaid-svg-Q0Cv61uzu3GVxhM0 text.actor>tspan{fill:#000;stroke:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .actor-line{stroke:grey}#mermaid-svg-Q0Cv61uzu3GVxhM0 .messageLine0{stroke-width:1.5;stroke-dasharray:none;stroke:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .messageLine1{stroke-width:1.5;stroke-dasharray:2, 2;stroke:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 #arrowhead path{fill:#333;stroke:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .sequenceNumber{fill:#fff}#mermaid-svg-Q0Cv61uzu3GVxhM0 #sequencenumber{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 #crosshead path{fill:#333;stroke:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .messageText{fill:#333;stroke:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .labelBox{stroke:#ccf;fill:#ECECFF}#mermaid-svg-Q0Cv61uzu3GVxhM0 .labelText,#mermaid-svg-Q0Cv61uzu3GVxhM0 .labelText>tspan{fill:#000;stroke:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .loopText,#mermaid-svg-Q0Cv61uzu3GVxhM0 .loopText>tspan{fill:#000;stroke:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .loopLine{stroke-width:2px;stroke-dasharray:2, 2;stroke:#ccf;fill:#ccf}#mermaid-svg-Q0Cv61uzu3GVxhM0 .note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-Q0Cv61uzu3GVxhM0 .noteText,#mermaid-svg-Q0Cv61uzu3GVxhM0 .noteText>tspan{fill:#000;stroke:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .activation0{fill:#f4f4f4;stroke:#666}#mermaid-svg-Q0Cv61uzu3GVxhM0 .activation1{fill:#f4f4f4;stroke:#666}#mermaid-svg-Q0Cv61uzu3GVxhM0 .activation2{fill:#f4f4f4;stroke:#666}#mermaid-svg-Q0Cv61uzu3GVxhM0 .mermaid-main-font{font-family:"trebuchet ms", verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .section{stroke:none;opacity:0.2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .section0{fill:rgba(102,102,255,0.49)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .section2{fill:#fff400}#mermaid-svg-Q0Cv61uzu3GVxhM0 .section1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .section3{fill:#fff;opacity:0.2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .sectionTitle0{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .sectionTitle1{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .sectionTitle2{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .sectionTitle3{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .sectionTitle{text-anchor:start;font-size:11px;text-height:14px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .grid .tick{stroke:#d3d3d3;opacity:0.8;shape-rendering:crispEdges}#mermaid-svg-Q0Cv61uzu3GVxhM0 .grid .tick text{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .grid path{stroke-width:0}#mermaid-svg-Q0Cv61uzu3GVxhM0 .today{fill:none;stroke:red;stroke-width:2px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .task{stroke-width:2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText{text-anchor:middle;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText:not([font-size]){font-size:11px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutsideRight{fill:#000;text-anchor:start;font-size:11px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutsideLeft{fill:#000;text-anchor:end;font-size:11px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .task.clickable{cursor:pointer}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutsideLeft.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutsideRight.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskText3{fill:#fff}#mermaid-svg-Q0Cv61uzu3GVxhM0 .task0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .task1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .task2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .task3{fill:#8a90dd;stroke:#534fbc}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutside0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutside2{fill:#000}#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutside1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .taskTextOutside3{fill:#000}#mermaid-svg-Q0Cv61uzu3GVxhM0 .active0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .active1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .active2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .active3{fill:#bfc7ff;stroke:#534fbc}#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeText0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeText1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeText2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeText3{fill:#000 !important}#mermaid-svg-Q0Cv61uzu3GVxhM0 .done0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .done1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .done2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .done3{stroke:grey;fill:#d3d3d3;stroke-width:2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneText0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneText1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneText2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneText3{fill:#000 !important}#mermaid-svg-Q0Cv61uzu3GVxhM0 .crit0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .crit1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .crit2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .crit3{stroke:#f88;fill:red;stroke-width:2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCrit0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCrit1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCrit2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCrit3{stroke:#f88;fill:#bfc7ff;stroke-width:2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCrit0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCrit1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCrit2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCrit3{stroke:#f88;fill:#d3d3d3;stroke-width:2;cursor:pointer;shape-rendering:crispEdges}#mermaid-svg-Q0Cv61uzu3GVxhM0 .milestone{transform:rotate(45deg) scale(0.8, 0.8)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .milestoneText{font-style:italic}#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCritText0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCritText1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCritText2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .doneCritText3{fill:#000 !important}#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCritText0,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCritText1,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCritText2,#mermaid-svg-Q0Cv61uzu3GVxhM0 .activeCritText3{fill:#000 !important}#mermaid-svg-Q0Cv61uzu3GVxhM0 .titleText{text-anchor:middle;font-size:18px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.classGroup text{fill:#9370db;stroke:none;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:10px}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.classGroup text .title{font-weight:bolder}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.clickable{cursor:pointer}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.classGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.classGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 .classLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.5}#mermaid-svg-Q0Cv61uzu3GVxhM0 .classLabel .label{fill:#9370db;font-size:10px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .relation{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .dashed-line{stroke-dasharray:3}#mermaid-svg-Q0Cv61uzu3GVxhM0 #compositionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #compositionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #aggregationStart{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #aggregationEnd{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #dependencyStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #dependencyEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #extensionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 #extensionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 .commit-id,#mermaid-svg-Q0Cv61uzu3GVxhM0 .commit-msg,#mermaid-svg-Q0Cv61uzu3GVxhM0 .branch-label{fill:lightgrey;color:lightgrey;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .pieTitleText{text-anchor:middle;font-size:25px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .slice{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.stateGroup text{fill:#9370db;stroke:none;font-size:10px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.stateGroup text{fill:#9370db;fill:#333;stroke:none;font-size:10px}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.statediagram-cluster .cluster-label text{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.stateGroup .state-title{font-weight:bolder;fill:#000}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.stateGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-Q0Cv61uzu3GVxhM0 g.stateGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-Q0Cv61uzu3GVxhM0 .transition{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-Q0Cv61uzu3GVxhM0 .stateGroup .composit{fill:white;border-bottom:1px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .stateGroup .alt-composit{fill:#e0e0e0;border-bottom:1px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .state-note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-Q0Cv61uzu3GVxhM0 .state-note text{fill:black;stroke:none;font-size:10px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .stateLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.7}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edgeLabel text{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .stateLabel text{fill:#000;font-size:10px;font-weight:bold;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-Q0Cv61uzu3GVxhM0 .node circle.state-start{fill:black;stroke:black}#mermaid-svg-Q0Cv61uzu3GVxhM0 .node circle.state-end{fill:black;stroke:white;stroke-width:1.5}#mermaid-svg-Q0Cv61uzu3GVxhM0 #statediagram-barbEnd{fill:#9370db}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-cluster rect{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-cluster rect.outer{rx:5px;ry:5px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-state .divider{stroke:#9370db}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-state .title-state{rx:5px;ry:5px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-cluster.statediagram-cluster .inner{fill:white}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-cluster.statediagram-cluster-alt .inner{fill:#e0e0e0}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-cluster .inner{rx:0;ry:0}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-state rect.basic{rx:5px;ry:5px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-state rect.divider{stroke-dasharray:10,10;fill:#efefef}#mermaid-svg-Q0Cv61uzu3GVxhM0 .note-edge{stroke-dasharray:5}#mermaid-svg-Q0Cv61uzu3GVxhM0 .statediagram-note rect{fill:#fff5ad;stroke:#aa3;stroke-width:1px;rx:0;ry:0}:root{--mermaid-font-family: '"trebuchet ms", verdana, arial';--mermaid-font-family: "Comic Sans MS", "Comic Sans", cursive}#mermaid-svg-Q0Cv61uzu3GVxhM0 .error-icon{fill:#522}#mermaid-svg-Q0Cv61uzu3GVxhM0 .error-text{fill:#522;stroke:#522}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edge-thickness-normal{stroke-width:2px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edge-thickness-thick{stroke-width:3.5px}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edge-pattern-solid{stroke-dasharray:0}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-Q0Cv61uzu3GVxhM0 .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-Q0Cv61uzu3GVxhM0 .marker{fill:#333}#mermaid-svg-Q0Cv61uzu3GVxhM0 .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-Q0Cv61uzu3GVxhM0 {color: rgba(0, 0, 0, 0.75);font: ;}

整体实现

比特逆序

旋转因子

三大循环

算法原理

X(k)=∑n=0N−1x(n)×e−j2πNknX(k) = \sum_{n=0}^{N-1} x(n)\times e^{-j\frac{2\pi}{N}kn} \\ X(k)=n=0∑N−1x(n)×e−jN2πkn
在将信号进行补零，变为偶数序列的时候
0<=k<=N/2X(k)=X1(k)+Wnk×X2(k)0<=k<=N/2 \\ X(k) = X_1(k) + W_n^k\times X_2(k) 0<=k<=N/2X(k)=X1(k)+Wnk×X2(k)
再利用WNkW_N^kWNk的可约性质，可以得到
0<=k<=N/2X(k+N2)=X1(k)−Wnk×X2(k)0<=k<=N/2 \\ X(k+\frac{N}{2}) = X_1(k) - W_n^k\times X_2(k) 0<=k<=N/2X(k+2N)=X1(k)−Wnk×X2(k)

一些重要的关系

关于蝶群和蝶形运算

对于图中表的红色方框的就是一个蝶群，可以看到一个蝶群中可能包含有多个蝶形，左边第一个蝶群中就是包含了一个蝶形，而右边包含了两个蝶形。
经过分析可以知道，如果以左边的输入序列为第0级，而最右边为第M级（假设序列的长度为2的M次幂），则可以知道第M级只有一个蝶群，这个蝶群中含有2(m−1)2^(m-1)2(m−1)个蝶形和旋转因子。
经过分析可以看到第i级，有2M−i2^{M-i}2M−i个蝶群，比如以上图为例，第一级有23−1=42^{3-1}=423−1=4个蝶群。

关于旋转因子的分析

我们知道对于每一级分解之后，关于$W_N^k$中的N的值是不断变化的，从最后一级来分析。

N=2MWNk=e−j2π2MnkN = 2^M \\ W_N^k = e^{-j\frac{2\pi}{2^M}nk} N=2MWNk=e−j2M2πnk
对于倒数第二级
N=2(M−1)WNk=e−j2π2(M−1)nkN = 2^(M-1) \\ W_N^k = e^{-j\frac{2\pi}{2^(M-1)}nk} N=2(M−1)WNk=e−j2(M−1)2πnk

对于第i级
N=2(M−(M−i))WNk=e−j2π2(M(M−i))nk=e−j2π2iN = 2^(M-(M-i)) \\ W_N^k = e^{-j\frac{2\pi}{2^(M(M-i))}nk} \\ = e^{-j\frac{2\pi}{2^i}} N=2(M−(M−i))WNk=e−j2(M(M−i))2πnk=e−j2i2π

于是可以得到一个普遍的规律，对于第i级的旋转因子为
WNk=e−j2π2iW_N^k = e^{-j\frac{2\pi}{2^i}} WNk=e−j2i2π

编程主要使用到了三大循环

最外层的循环：级数的循环
中间的循环：旋转因子的个数（也是每一个蝶群中，蝶形的个数）
最内层的循环：一个旋转因子需要给多少个蝶群使用。

void fft(complexNumber* sequence, int order, int length, complexNumber * result)
{
int i, j, k;
copyComplex(sequence, length, result);
//THE FIRST LOOP
for(i=0; i<order; i++)
{int total_factors;complexNumber u = {1, 0};total_factors = pow(2, i);factor = getFactor(i+1);//printf("The %d's level loop\n", i);//THE NUMBER OF DIFFERENT FACTORS = THE LENGTH OF TWO GROUPSfor(j=0; j<total_factors; j++){int total_groups, index;total_groups = pow(2, order-i-1);  //THE NUMBER OF DIFFERENT GROUPSfor(index=0, k=j; index<total_groups;k+=total_factors*2, index++) {//COMPUTER THE DFTresult[k] = complex_add(*(sequence+k) ,complex_mul(u, *(sequence+k+total_factors)));result[k+total_factors] = complex_add(*(sequence+k) , reverse_complex(complex_mul(u, *(sequence+k+total_factors))));}//printf("The value of u is: \n");//showComplex(u);u = complex_mul(u, factor);}copyComplex(result, length, sequence);
}

}

这里仍然有很多需要注意的细节，比如关于逆序的时候，我们需要首先判断需要使用多少位宽的空间来存放数据，然后才能进行逆序，如果直接使用一个固定的位宽来进行逆序，那么除非你将所有的输入的序列继续补零到固定的值，然后再对这个整体的数据进行逆序，否则就是错误的，但是如果补零到固定的个数，那么就对造成浪费，而且fft的点数也就确定了。
**还有一个需要注意的点是，相邻两个蝶群的距离是多少，为什么需要这个参数呢？因为只有得到这个参数之后，我们才能知道同一个旋转因子，下一次需要的位置, 2i2^i2i，就是总共旋转因子个数（一个蝶群中蝶形的个数）的二倍。

结果展示

使用matlab的标准结果

源码，在上传的资源中。有任何问题欢迎交流。
源码地址