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