信号与系统Signals & SystemsElectronic Technology Teaching & Research Section5-6 离散时间傅里叶变换-DTFT一、离散时间傅里叶变换的定义设离散时间序列x(n)的z变换=nnznxzX )()(单位圆被包含在它的收敛域之内。于是=nnjezjenxzXeXj)(|)()(定义为序列x(n)的离散时间傅里叶变换（DTFT）。记为=nnjjenxnxDTFTeX)()()(信号与系统Signals & SystemsElectronic Technology Teaching & Research Section由离散时间序列x(n)的反z变换=CndzzzXjnx1)(21)(由于单位圆在X(z)的收敛域之内，以上围线积分可沿单位圆上进行。于是=11)(21)(zndzzzXjnx=jnjjdeeeXj)1()(21=djeeeXjjnjj )1()(21=deeXnjj)(21记为=deeXeXIDTFTnxnjjj)(21)()(信号与系统Signals & SystemsElectronic Technology Teaching & Research Section于是，我们得到一对变换关系：=nnjjenxnxDTFTeX)()()( -DTFT变换式=deeXeXIDTFTnxnjjj)(21)()(-DTFT反变换式)()(jDTFTeXnx 记为物理含义：1 DTFT是将序列x(n)分解为不同角频率的复指数序列ejn的组合;2 X(ej)是不同分量的复振幅的相对大小，习惯上，称X(ej)是序列x(n)的频谱; 3 X(ej)是连续的，并且以2为周期信号与系统Signals & SystemsElectronic Technology Teaching & Research Section二、离散时间傅里叶变换的举例1、单边指数序列)()( nuanxn= 1RezImzj1a当|a|ajDTFTnaenua11)(即信号与系统Signals & SystemsElectronic Technology Teaching & Research Section2、双边指数序列nanx =)(1a于是=nnjjenxeX)()( =+=01nnjnnnjneaea其中=11nnjnnnjneaeajjaeae=1所以jjjjaeaeaeeX+=111)(22cos211aaa+=3、矩形窗序列)()()()( NnununRnxN=n)()(4nRnx =0123 4 51=nnjjenxeX)()(=10Nnnje信号与系统Signals & SystemsElectronic Technology Teaching & Research SectionjNjjeeeX=11)(2/2/2/2/2/2/)()(jjjNjNjNjeeeeee=21)2sin()2sin( =NjeN)2sin()2sin()(NeXj=)()( jjeeX=+=2sin2sinarg21)(NN)(jeX24=N0)(204343n)()(4nRnx =0123 4 51信号与系统Signals & SystemsElectronic Technology Teaching & Research Section三、离散时间傅里叶变换的基本性质1、周期性)()()2( =jjeXeX即序列是时域离散的，其离散时间傅里叶变换是以2为周期的周期信号。注意，连续时间周期信号，其连续时间傅里叶变换是离散的。例如：单边指数序列jnjaenuaDTFTeX=11)()()2(1111 =jjaeae 211jjeaem=2、线性设)()(jiDTFTieXnx 信号与系统Signals & SystemsElectronic Technology Teaching & Research Section则ijiiDTFTiiieXCnxC )()(例如：双边指数序列)()1()( nuanuanxnn+=1a则)()1()( nuaDTFTnuaDTFTeXnnj+=jjjaeaeae+=11122cos211aaa+=3、时移与频移性设)()(jDTFTeXnx 则有mjjDTFTeeXmnx )()()()()(00 jDTFTnjeXenx信号与系统Signals & SystemsElectronic Technology Teaching & Research Section例题：设矩形窗序列RN(n)的宽度N为奇数，21)2sin()2sin()(NjDTFTNeNnRn)()(5nRnx =0123 4 51)(jeX25=N0)(205454我们已知则RN(n)左移( N-1)/2后，是一个偶对称的序列,n)21(+Nnx1233112根据时移性)2sin()2sin()21(NNnRDTFTN+信号与系统Signals & SystemsElectronic Technology Teaching & Research Section)2sin()2sin()21(NNnRDTFTN+)(jeX25=N0)(20)(jeX25=N0因为，此时序列是一偶对称信号，与连续时间傅氏变换相同，其变换应是纯实函数。变换的波形如图所示。n)21(+Nnx1233112离散时间信号的傅立叶变换是以2为周期的连续函数，其幅度函数的波形是以偶对称的，相位函数是奇对称的。信号与系统Signals & SystemsElectronic Technology Teaching & Research Section例题：设)()(jDTFTeXnx 则由频移性)()()()1()( =jDTFTjnneXnxenx)(nxn0 123 4)(jeX022)(1nxn01234)(1jeX022信号与系统Signals & SystemsElectronic Technology Teaching & Research Section4、共轭与反褶设)()(jDTFTeXnx 则有)()(* jDTFTeXnx)()(jDTFTeXnx所以有)()(21)()(21)(Re* jjDTFTeXeXnxnxnx+=)()(21)()(21)(Im* jjDTFTeXeXjnxnxjnxj=即序列实部的离散时间傅里叶变换是序列离散时间傅里叶变换的共轭对称分量，虚部的离散时间傅里叶变换是序列离散时间傅里叶变换的共轭反对称分量。（推导另一个结论）信号与系统Signals & SystemsElectronic Technology Teaching & Research Section5、奇、偶、虚、实性设)()()()()()( IRjDTFTirjXXeXnjxnxnx +=+=)()( jjeeX=当x(n)是实序列，即)()(*nxnx =则)()(* jjeXeX=即实序列的离散时间傅里叶变换，实部是偶对称的，虚部是奇对称的，模是偶对称的，相位是奇对称的。)()()()()()()()(=+=jjIRIRjjeeXjXXjXXeeX当x(n)是实偶序列，即)()()(*nxnxnx =信号与系统Signals & SystemsElectronic Technology Teaching & Research Section则)()()(* jjjeXeXeX=)()()()()()( +=+IRIRIRjXXjXXjXX即0)( =IX当x(n)是实奇序列，即)()()(*nxnxnx =则)()()(* jjjeXeXeX=)()()()()()( =+IRIRIRjXXjXXjXX即0)( =RX即实偶序列的离散时间傅里叶变换，是实偶对称的；实奇序列，其离散时间傅里叶变换是纯虚且奇对称的。信号与系统Signals & SystemsElectronic Technology Teaching & Research Section同样可求，其中的奇分量)()(21)()(21)( jjDTFTeeXeXnxnxnx+=)(Im)()(21)()(21)( jjjDTFToeXjeXeXnxnxnx =当x(n)是实序列，即)()(*nxnx =则其中的偶分量=+)()(21 jjeXeX )()()()(21 +IRIRjXXjXX)(RX=)(RejeX=信号与系统Signals & SystemsElectronic Technology Teaching & Research Section设)()(jDTFTeXnx 则有dedXjnnxjDTFT)()( 6、频域微分性(序列线性加权）例如：已知jDTFTnaenua11)(则有11)(jDTFTnaeddjnuna2)1(jjaeae=7、卷积定理设)()(11jDTFTeXnx )()(22jDTFTeXnx 信号与系统Signals & SystemsElectronic Technology Teaching & Research Section则有)()()()(2121 jjDTFTeXeXnxnx )()(21)()(2121jjDTFTeXeXnxnx 这里的卷积表示式=deXeXeXeXjjjj)()()()()(2121=deXeXjj)()()(12称为周期卷积。参加卷积的两信号均是以2为周期的周期信号，卷积的积分是在2区间上的积分，卷积后的结果，仍然是以2为周期的周期信号。有关它的运算，在数字信号处理讨论。信号与系统Signals & SystemsElectronic Technology Teaching & Research Section设)()(jDTFTeXnx 则有=deXnxjn22)