Chapter 3 Finite Length Discrete Transforms
Chapter 3 Finite Length Discrete Transforms
Introduce Often, in practice, it is convenient to map a finite-length sequence from the time domain into a finite-length sequence of the same length in a different domain and vice-versa e Such transformations are usually collectively called finite-length transforms
Introduce ⚫ Often, in practice, it is convenient to map a finite-length sequence from the time domain into a finite-length sequence of the same length in a different domain, and vice-versa. ⚫ Such transformations are usually collectively called finite-length transforms
Introduce o In some applications, a very long length time domain sequence is broken up into a set of short-length time-domain sequences and a finite- length transform is applied to each short-length sequence
Introduce ⚫ In some applications, a very long length timedomain sequence is broken up into a set of short-length time-domain sequences and a finitelength transform is applied to each short-length sequence
Introduce o The transformed sequences are next processed in the transform domain, and their time-domain equivalents are generated by applying the inverse transform. The processed shorted-length sequences are then grouped together appropriately to develop the final long-length sequence
Introduce ⚫ The transformed sequences are next processed in the transform domain, and their time-domain equivalents are generated by applying the inverse transform. The processed shorted-length sequences are then grouped together appropriately to develop the final long-length sequence
Orthogonal Transform o a general form of the orthogonal transform pair is of the form X[k]=∑x(n)[kn0≤k≤N-1 Analysis equation x[]=∑X[小[kn0≤n≤N k=0 Basis synthesis sequences equation
Orthogonal Transform ⚫ A general form of the orthogonal transform pair is of the form ( ) 1 * 0 , , 0 1 N n X k x n k n k N − = = − 1 0 1 , , 0 1 N k x n X k k n n N N − = = − Analysis equation synthesis equation Basis sequences
Orthogonal Transform o The basis sequences satisfy the condition 「1,1=k ∑vk,n]v[,n n=0 0.l≠k
Orthogonal Transform ⚫ The basis sequences satisfy the condition 1 * 0 1 1, , , 0, N n l k k n l n N l k − = = =
Orthogonal Transform o An important consequence of the orthogonality of the basis sequence is energy preservation property of the transform ∑ 可=∑xi N k=0
Orthogonal Transform ⚫ An important consequence of the orthogonality of the basis sequence is energy preservation property of the transform 1 1 2 2 0 0 1 N N n k x n X k N − − = = =
Discrete Fourier Transform ● Definition ● Matrix relations o DFT Computation Using MATLAB o Relation between dTFT and dFt and their Inverses
Discrete Fourier Transform ⚫ Definition ⚫ Matrix Relations ⚫ DFT Computation Using MATLAB ⚫ Relation between DTFT and DFT and their inverses
Discrete Fourier transform o In this section we define the discrete fourier transform, usually known as the DFT, and develop the inverse transformation, often abbreviated as DT
Discrete Fourier transform ⚫ In this section, we define the discrete Fourier transform, usually known as the DFT, and develop the inverse transformation, often abbreviated as IDFT
1 Definition The discrete Fourier transform(DFt) of length-N time domain sequence xn] is defined by X[]=∑x[e2xb,0≤k≤N-1 The basis sequence is plk.nl=e 2Tkn/n Which are complex exponential sequences commonly used notation 2丌/N
1. Definition ⚫ The discrete Fourier transform (DFT) of length-N time domain sequence x[n] is defined by 1 2 / 0 , 0 1 N j kn N n X k x n e k N − − = = − ⚫ The basis sequence is ,which are complex exponential sequences commonly used notation 2 / , j kn N k n e = j N 2 / W e N − =