倒频谱 （cepstrum ），顾名思义，就是将频谱 （spectrum）的英文前四个字母反过来写。倒频谱是为了某些时候，为了计算方便，将原来信号的频谱先转成类似分贝 的单位，再作逆傅里叶变换 ，把它视为一种新的讯号做处理。倒频谱有复数 倒频谱，及实数 倒频谱。
倒频谱的范例 倒频谱被定义在1963的论文（Bogert等）。定义如下：
字义：倒频谱（信号）是信号频谱取对数的傅立叶变换后的新频谱（信号），有时候会称频谱的倒频谱。
数学上：信号的倒频谱 = IFT ( log ( | FT (信号) | ) + j2πm )（m为实数）
演算法：信号 -> 傅立叶变换 -> 取绝对值 -> 取对数 -> 相位展开 -> 逆傅立叶变换 -> 倒频谱 复数倒频谱拥有频谱大小跟相位的资讯，实数倒频谱只有频谱大小的资讯，各有各的不同应用。
倒频谱可以被视为在不同频带上变化速率的资讯，倒频谱一开始被发明在地震 或炸弹 产生的地震回音，现今也被使用在分析雷达 讯号，以及讯号处理 等问题。
自相关倒频谱(autocepstrum)被定义为倒频谱的自相关性，自相关倒频谱有时在分析处理回传讯号时比倒频谱还准确。
倒频谱在处理人声 讯号以及音乐讯号有非常好的效果，例如梅尔频率倒频谱(Mel-Frequency Cepstrum)，用来做声音的辨认，侦测音高等。近年来梅耳倒频谱也被应用在音乐资讯的回复。
倒频谱在声学 中可以将声带 震动的影响去除。
倒频谱用在处理多路径问题时(如声波 的回音 、电磁波 的折、反射等)，如果将其他路径干扰 视为杂讯 ，为了消除杂讯，利用倒频谱，不需测量每条多路径的延迟时间，可以利用传送多次信号，观察其他路径在倒频谱上的效果，并且加以滤除。
语音大致上是由音高、声带 脉冲 、声门波形所组成，我们可以利用倒频谱将这三种元素在倒频域上分开，以利于做语音 讯号的分析。
倒频谱的微分适用于影像处理上的图形辨认(pattern recognition)。
倒频谱与同型声音理论(homomorphic sound theory)有关。
频谱图上的独立变数是频率 ，而倒频谱图上的独立变数为倒频率(quefrency)，倒频率是一种时间的度量单位 。举个例子，声音讯号取样速率等于44100赫兹 ，在倒频谱上有个很大的值在倒频率等于100，代表实际上在44100/100=441赫兹 有很大的值，这值出现在倒频谱上因为频谱上周期 性出现，而频谱上出现的周期与倒频谱很大的值出现的位置有关。
滤波器 (filter)常使用在频谱上，用来保存或删除我们所要或不要的资讯，经过上面的许多讨论，不难猜到，倒滤波器(lifter)就是在倒频谱上所使用的滤波器。低通的倒滤波器跟低通滤波器有点类似，它可以借由在倒频谱上乘以一个window系数，使倒频谱上的高倒频率被压抑，如此依来，当信号转回时域 空间时会变成一个较平滑的信号。
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先对信号做Z转换 , 并整理一下系数, 让他变成下面的形式
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{\displaystyle {\widehat {X}}\left(Z\right)=logX\left(Z\right)=\log A+r\log Z+\sum _{k=1}^{m_{i}}\log(1-{a_{k}}{Z^{-1}})+\sum _{k=1}^{m_{0}}\log(1-{b_{k}}Z)-\sum _{k=1}^{P_{i}}\log(1-{c_{k}}{Z^{-1}})-\sum _{k=1}^{P_{0}}\log(1-{d_{k}}Z)}
Z转换 所得到的系数, 我们可以利用泰勒展开得到Z的反转换
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无限脉冲响应 )有限脉冲响应 )的情况,
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最小相位 (minimum phase) i.e.
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1. 复数倒频谱至少以
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梅尔频率倒频谱 是倒频谱的一种应用，梅尔频率倒频谱常应用在声音讯号处理，对于声音讯号处理比倒频谱更接近人耳对声音的分析特性，而梅尔频率倒频谱与倒频谱的差别在于:
梅尔频率倒频谱的频带分析是根据人耳听觉特性所设计，人耳对于频率 的分辨能力，是由频率的"比值"决定，也就是说，人耳对200赫兹 和300赫兹之间的差别与2000赫兹和3000赫兹之间的差别是相同的。
梅尔频率倒频谱是针对讯号的能量 取对数 ，而倒频谱是针对讯号原始在频谱上的值取对数 。
梅尔频率倒频谱是使用离散馀弦转换 ，倒频谱是用离散傅立叶变换 。
梅尔频率倒频谱系数足够描述语音 的特征。 梅尔频率倒频谱系数 (MFCCs)的推导步骤：
将信号做傅立叶变换
频谱上的值取绝对值再平方成为能量，在乘上频谱上对应的梅尔频率倒频谱三角重叠窗(window)的系数。
对每个梅尔频率取对数 。
作离散馀弦转换 。
求得梅尔频率倒频谱系数。
梅尔频率倒频谱系数常利用在辨认语音技术上，例如辨认电话中说话的人的身份。
利用每种乐风 、或乐器 在梅尔频域上有不同特性来分析音乐 的种类与类型，并且可以加以分类。
梅尔频率倒频谱系数很容易被外来的杂讯所破坏，因此有些研究结果指出，在求梅尔频率倒频谱系数时，在作离散馀弦转换前，提升适当的能量(大约2或3倍)，以减少杂讯在低能量成份的影响。
倒频谱领域上的一项重要的特性为二信号卷积 之产生，其产生之程序为二倒频谱值(cepstra)之相加：
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(1)微分倒频谱在shift和scaling时，结果不改变。
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复数倒频谱的衰减率反比于n
微分倒频谱的衰减率下降
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{\displaystyle \therefore {\widehat {x}}_{d}(n+1)=n{\widehat {c}}(n)\varpropto n{\frac {1}{n}}=1}
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{\displaystyle x[n]\quad {\stackrel {Ztransform}{\longrightarrow }}\quad X(Z)\quad {\stackrel {log}{\longrightarrow }}\quad {\widehat {X}}(Z)\quad {\stackrel {Z^{-1}}{\longrightarrow }}\quad {\widehat {x}}[n]}
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{\displaystyle {\widehat {x}}[n]=\sum _{n=0}^{N}-{\frac {-0.5^{n}}{n}},n>0}
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{\displaystyle {\widehat {x}}[0]=1}
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{\displaystyle {\widehat {x}}[n]\quad {\stackrel {Ztransform}{\longrightarrow }}\quad {\widehat {X}}(Z)\quad {\stackrel {exp}{\longrightarrow }}\quad {X}(Z)\quad {\stackrel {Z^{-1}}{\longrightarrow }}\quad {x}[n]}
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{\displaystyle x[n]={\begin{cases}{\frac {1}{n!}},n\geq 0\\0,otherwise\\\end{cases}}}
Suppose that an IIR filter is
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{\displaystyle H(Z)={\frac {2z^{3}-4z^{2}-z+2}{2z^{2}-2z+1}}}
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{\displaystyle x[n]\quad {\stackrel {Ztransform}{\longrightarrow }}\quad X(Z)\quad {\stackrel {log}{\longrightarrow }}\quad {\widehat {X}}(Z)\quad {\stackrel {Z^{-1}}{\longrightarrow }}\quad {\widehat {x}}[n]}
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{\displaystyle H(Z)={\frac {(-2)(z)(z-{\frac {\sqrt {2}}{2}}z^{-1})(z+{\frac {\sqrt {2}}{2}}z^{-1})(1-{\frac {1}{2}}z)}{(1-{\frac {1+j}{2}}z^{-1})(1-{\frac {1-j}{2}}z^{-1})}}}
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{\displaystyle {\widehat {H}}(Z)=log(-2)+3log(z)+log(1\pm {\frac {\sqrt {2}}{2}}z^{-1})+log(1-{\frac {1}{2}}z)-log(1-{\frac {1\pm j}{2}}z^{-1})}
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{\displaystyle {\widehat {h}}[n]={\begin{cases}log(-2),n=0\\\displaystyle {-{\frac {{({\frac {\sqrt {2}}{2}})}^{n}+{({\frac {-{\sqrt {2}}}{2}})}^{n}}{n}}+{\frac {{({\frac {1+j}{2}})}^{n}+{({\frac {1-j}{2}})}^{n}}{n}},n>0}\\\displaystyle {\frac {{({\frac {1}{2}})}^{-n}}{n}},n<0\\\end{cases}}}
{\displaystyle }
B. P. Bogert, M. J. R. Healy, and J. W. Tukey : "The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance , cross-cepstrum, and saphe cracking". Proceedings of the Symposium on Time Series Analysis (M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley, 1963.
D. G. Childers, D. P. Skinner, R. C. Kemerait, "The Cepstrum: A Guide to Processing （页面存档备份 ，存于互联网档案馆 ）," Proceedings of the IEEE , Vol. 65, No. 10, October 1977, pp. 1428-1443.
Jian-Jiun Ding, Advanced Digital Signal Processing class note,the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2008
Jian-Jiun Ding, Advanced Digital Signal Processing class note,the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2024
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