(复)频域变换

(复)频域变换定义

正变换 逆变换
傅里叶级数 a k = 1 T T f ( t ) e j k ω 0 t d t f ( t ) = k = a k e j k ω 0 t
傅里叶级数的帕斯瓦关系 1 T T | f ( t ) | 2 d t = k = | a k | 2
傅里叶变换 F ( ω ) = x f ( t ) e j ω t d t f ( t ) = 1 2 π F ( ω ) e j ω t d ω
傅里叶变换的帕斯瓦关系 | f ( t ) | 2 d t = 1 2 π | F ( ω ) | 2 d ω
离散傅里叶级数 a k = 1 N n =< N > f [ n ] e j k Ω 0 n f [ n ] = k =< N > a k e j k Ω 0 n
离散傅里叶级数的帕斯瓦关系 1 N n = 0 N 1 | f [ n ] | 2 = k = 0 N 1 | a k | 2
离散时间的傅里叶变换 F ( Ω ) = n = f [ n ] e j Ω n f [ n ] = 1 2 π 2 π F ( Ω ) e j Ω n d Ω
离散时间的傅里叶变换的帕斯瓦关系 n = | f [ n ] | 2 = 1 2 π 0 2 π | F ( e j Ω ) | 2 d Ω

(复)频域变换性质

性质 傅里叶变换 拉普拉斯变换 z变换
时域 频域 时域 复频域 时域 复频域
线性 a 1 f 1 ( t ) + a 2 f 2 ( t ) a 1 F 1 ( ω ) + a 2 F 2 ( ω ) a 1 f 1 ( t ) + a 2 f 2 ( t ) a 1 F 1 ( s ) + a 2 F 2 ( s ) a 1 f 1 [ n ] + a 2 f 2 [ n ] a 1 F 1 ( z ) + a 2 F 2 ( z )
尺度变换 f ( a t ) 1 | a | F ( ω a ) f ( a t ) 1 | a | F ( s a ) f [ n ] F ( 1 z )
F ( t ) 2 π f ( ω ) f 1 [ n ] = { f [ n N ] , n = k N 0 , n k N F ( z N )
共轭对称性 f ( t ) F ( ω )
f ( t ) 是实函数 F ( ω ) = F ( ω ) R e { F ( ω ) } = R e { F ( ω ) } I m { F ( ω ) } = I m { F ( ω ) } f ( t ) 是实函数 F ( s ) = F ( s ) f [ z ] F ( z ) = F ( z )
f e ( t ) = 1 2 [ f ( t ) + f ( t ) ]
f ( t ) 是实函数
R e { F ( ω ) }
f o ( t ) = 1 2 [ f ( t ) f ( t ) ]
f ( t )\ 是实函数
j I m { F ( ω ) }
f ( t ) 是实偶函数 F ( ω ) 是实偶函数
f ( t ) 是实奇函数 F ( ω ) 是虚奇函数
时移 f ( t t 0 ) e j ω t 0 F ( ω ) f ( t t 0 ) e s t 0 F ( s ) f [ n n 0 ] z n 0 F ( z )
(复)频移 e j ω 0 t f ( t ) F ( ω ω 0 ) e s 0 t f ( t ) F ( s s 0 ) z 0 n f [ n ] F ( z z 0 )
时域微分(差分) d f ( t ) d t j ω F ( ω ) d f ( t ) d t s F ( s ) f [ n ] f [ n 1 ] ( 1 z 1 ) F ( z )
单边时域微分(差分) f ( n ) ( t ) s n F u ( s ) s n 1 f ( 0 ) s n 2 f ( 0 ) f ( n 1 ) ( 0 ) f [ n n 0 ] z n 0 F u ( z ) + f [ n 0 ] + z 1 f [ n 0 + 1 ] + + z n 0 + 1 f [ 1 ]
f [ n + n 0 ] z n 0 F u ( z ) z n 0 f [ 0 ] z n 0 1 f [ 1 ] z f [ n 0 1 ]
时域积分(累加) t f ( τ ) d τ F ( ω ) j ω + π F ( 0 ) δ ( ω ) t f ( τ ) d τ 1 s F ( s ) m = n f [ m ] 1 1 z 1 F ( z )
(复)频域微分 1 1 z 1 F ( z ) d F ( ω ) d ω t f ( t ) d F ( s ) d s n f [ n ] z d F ( z ) d z
(复)频域积分 f ( t ) j t + π f ( 0 ) δ ( t ) ω F ( θ ) d θ f ( t ) t s F ( s ) d s
时域卷积 f ( t ) h ( t ) F ( ω ) H ( ω ) f ( t ) h ( t ) F ( s ) H ( s ) f [ n ] h [ n ] F ( z ) H ( z )
频域卷积 f ( t ) h ( t ) 1 2 π F ( ω ) H ( ω )
初值 lim t 0 + f ( t ) = lim s s F ( s ) f [ n 0 ] = lim z [ z n 0 F ( z ) ]
终值 lim t f ( t ) = lim s 0 s F ( s ) f [ ] = lim z 1 [ ( 1 z 1 ) F ( z ) ]

常见函数的(复)频域变换

连续时间信号 傅里叶变换 连续时间信号 拉普拉斯变换 离散时间信号 z变换
δ ( t ) 1 δ ( t ) 1 δ [ n ] 1
δ ( t t 0 ) e j ω t δ ( t t 0 ) e s t δ [ n n 0 ] z n 0
δ ( n ) ( t ) ( j ω ) n δ ( n ) ( t ) s n
1 2 π δ ( ω )
u ( t ) 1 j ω + π δ ( ω ) u ( t ) / u ( t ) 1 s u [ n ] / u [ n 1 ] 1 1 z 1
e a t u ( t ) , R e { a } > 0 1 j ω + a e a t u ( t ) / e a t u ( t ) 1 s + a a n u [ n ] / a n u [ n 1 ] 1 1 a z 1
t e a t u ( t ) , R e { a } > 0 1 ( j ω + a ) 2 t e a t u ( t ) / t e a t u ( t ) 1 ( s + a ) 2 n a n u [ n ] / n a n u [ n 1 ] a z 1 ( 1 a z 1 ) 2
1 n ! t n e a t u ( t ) , R e { a } > 0 1 ( j ω + a ) n + 1 1 n ! t n e a t u ( t ) / 1 n ! t n e a t u ( t ) 1 ( s + a ) n + 1 n ( n 1 ) ( n k + 2 ) ( k 1 ) ! a n k + 1 u [ n ] z k + 1 ( 1 a z 1 ) k
1 n ! t n u ( t ) / 1 n ! t n u ( t ) 1 s n + 1 r 0 n e j Ω 0 n u [ n ] r 0 n e j Ω 0 n u [ n ]
u ( n ) ( t ) = 1 n ! t n u ( t ) 1 s n + 1 r 0 n e j Ω 0 n u [ n ] 1 1 ( r 0 e j Ω 0 ) z 1
cos ( ω 0 t ) π δ ( ω ω 0 ) + π δ ( ω + ω 0 ) cos ( ω 0 t ) u ( t ) s s 2 + ω 0 2 cos ( Ω 0 n ) u [ n ] 1 cos Ω 0 z 1 1 2 cos Ω 0 z 1 + z 2
e a t cos ( ω 0 t ) u ( t ) s + a ( s + a ) 2 + ω 0 2 r 0 n cos ( Ω 0 n ) u [ n ] 1 r 0 cos Ω 0 z 1 1 2 r 0 cos Ω 0 z 1 + r 0 2 z 2
sin ( ω 0 t ) j π δ ( ω + ω 0 ) j π δ ( ω ω 0 ) sin ( ω 0 t ) u ( t ) ω 0 s 2 + ω 0 2 sin ( Ω 0 n ) u [ n ] sin Ω 0 z 1 1 2 cos Ω 0 z 1 + z 2
e a t sin ( ω 0 t ) u ( t ) ω 0 ( s + a ) 2 + ω 0 2 r 0 n sin ( Ω 0 n ) u [ n ] r 0 sin Ω 0 z 1 1 2 r 0 cos Ω 0 z 1 + r 0 2 z 2
e a | t | , R e { a } > 0 2 a ω 2 + a 2 e a | t | 2 a s 2 a 2
g τ ( t ) = { 1 , | t | < τ 2 0 , | t | > τ 2 τ Sa ( ω τ 2 ) = 2 sin ( ω τ / 2 ) ω
sgn ( t ) = { 1 , t > 0 0 , t < 0 2 j ω
1 π t j sgn ( ω )
e j ω 0 t 2 π δ ( ω ω 0 )
k = a k e j k ω 0 t , ω 0 = 2 π T 2 π k = a k δ ( ω k ω 0 )

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