Lecture, week 10 The Discrete Fourier Transform II Week 10, INF3190/4190 Andreas Austeng Department of Informatics, University of Oslo October 2019 AA, IN3190/4190 (Ifi/UiO) Lecture, week 10 Oct. 2019 1 / 21 Outline Outline 1 Spectral smoothing Spectral smoothing Discrete time windows Window resolution 2 Using the DFT Linear convolution of long sequences Bandlimited interpolation AA, IN3190/4190 (Ifi/UiO) Lecture, week 10 Oct. 2019 2 / 21 Spectral smoothing Outline 1 Spectral smoothing Spectral smoothing Discrete time windows Window resolution 2 Using the DFT Linear convolution of long sequences Bandlimited interpolation AA, IN3190/4190 (Ifi/UiO) Lecture, week 10 Oct. 2019 3 / 21 Spectral smoothing Finite length data (I) If we define a window w [n]= ( 1, n 2 0, 1,..., N - 1 0 otherwise and a periodic signal x p [n], n 2 -1 ... 1, then the product x [n]= x p [n] w [n] describes a time limited version of x p [n]. Taking the DFT of x [n] might cause problems as: I Spectral leakage (when N 6= kT s , k 2 N, T s being the period of x p [n]) I Smoothing AA, IN3190/4190 (Ifi/UiO) Lecture, week 10 Oct. 2019 4 / 21
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Outline Lecture, week 10 The Discrete Fourier Transform II · Lecture, week 10 The Discrete Fourier Transform II Week 10, INF3190/4190 Andreas Austeng Department of Informatics, University
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and a periodic signal xp[n], n 2 �1 . . .1,then the product x [n] = xp[n] w [n]describes a time limited version of xp[n].Taking the DFT of x [n] might cause problems as:
I Spectral leakage (when N 6= kTs, k 2 N, Ts being the period of xp[n])I Smoothing
Using the DFT Linear convolution of long sequences
Convolution of long sequences
Given h[n] of length N1, and x [n] of length N2.1 Zero-pad h[n] and x [n] to length L � N1 + N2 � 1.2 Compute L-point DFT of h[n] and x [n].3 Perform multiplication: Y [k ] = H[k ] X [k ].4 Find inverse of Y [k ]! y [n] = h[n] ⇤ x [n].
Sometime, this is not practical:I Long sequences require much storage, large computational load, and long time (we
have to wait for last sample).Solution: Block-convolution. Two types: