Discrete Fourier Transform DFT JyhShing Roger Jang MIR

Discrete Fourier Transform (DFT) Jyh-Shing Roger Jang (張智星) MIR Lab (多媒體資訊檢索實驗室) CSIE, NTU (台灣大學 資訊 程系) jang@mirlab. org, http: //mirlab. org/jang

Discrete Fourier Transform z. Goal Due to sampling y. Decompose a given vector of discrete signals into sinusoidal components whose amplitudes can be view as energy distribution over a range of frequencies z. Applications y. Speech recognition y. Digital filtering y. Many more. . . -2 -

Decomposition z. How to decompose xn=x(n/fs), n=0~N-1 into a linear combination of sinusoidal functions? y. Magnitude spectrum y. Phase spectrum z. Remaining problems y. To determine no. of terms and their frequencies -3 -

How to Select Frequencies z. Signals of N points with sample rate fs y. Total duration d=N/fs (sec) y. Periods = d, d/2, d/3, … z. Freq. of sinusoids (bin frequencies) Total count = N/2+1 -4 -

No. of Terms for Decomposition z. Curve fitting for a parabola y# of unknowns = # of equations (= # of points) z. Curve fitting view for sinusoidal decomp. How many terms do we need if there are N points? N/2+1! -5 -

Example When N=4 -6 -

Example When N=8 -7 -

Sinusoidal Basis Functions (1/2) Can be converted into sin & cos Phase is necessary -8 -

Sinusoidal Basis Functions (2/2) -9 -

Decompose x(t) into Basis Functions n=0, 1, 2, …, N-1 # of terms = # of bin freq = N/2+1 # of fitting parameters = N -10 -

Decomposition into Basis Functions z. Overall expression: Amplitude Freq= Phase y. No. of parameters is 1+2*(N/2 -1)+1=N, which is equal to the no. of data points Exact solution is likely to exist. -11 -

Parameter Identification z. How to identify coefficients of the basis functions: y. Solving simultaneous linear equations y. Integration (which take advantage of the orthogonality of the basis functions) y. Fast Fourier transform (FFT) x. A fast algorithm with a complexity of O(N log N) -12 -

From Time-domain to Freq-domain z. Many animation over the web y 3 D animation Please post on FB if you find more. -13 -

Frequencies of Basis Functions z Due to Euler identity, we can express the k-th term shown on the right. Quiz! -14 -

Frequencies of Basis Functions z Plug in the simplified kth term: z Include the first and last terms: -15 -

Representations of DFT: Two-side z. Characteristics y. General representation y. If x is complex, then c is not conjugate symmetric y. MATLAB command: c=fft(x) Usually complex numbers -16 -

Representations of DFT: One-side z. Characteristics y. When x is real, c is conjugate symmetric and we only have to look at one-side of FFT result. y. MATLAB command: mag. Spec=fft. Oneside(x) Real In SAP toolbox Complex conjugate -17 -

Representations of DFT: One-side z. Formulas -18 -

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Example: Conjugate Symmetric z Conjugate symmetric of DFT for real x yfft. Sym 01. m -20 -

Example: Two-side FFT z Two-side FFT of a pure sinusoid at one of the bin freqency yfft 01. m -21 -

Example: Two-side FFT z Two-side FFT of a pure sinusoid NOT at one of the bin frequencies yfft 02. m -22 -

Example: One-side FFT z One-side FFT of a pure sinusoid NOT at one of the bin frequencies yfft 03. m -23 -

Example: One-side FFT z One-side FFT of a frame of audio signals yfft 04. m -24 -

Example: FFT for Data Compression z Use partial coefficients to reconstruct the original signals yfft. Approximate 01. m y. High-frequency components are not so important -25 -

Example: FFT for Data Compression z Use low-freq components to reconstruct a frame yfft. Approximate 01. m -26 -

Example: FFT for Periodic Signals Quiz! z FFT on periodic signals yfft. Repeat 01. m y. Useful for pitch tracking y. Interpretation of harmonics from viewpoints of x. Approximation by basis functions x. Integration for obtaining the coefficients -27 -

Why Harmonic Structures? -28 -

Example: Zero-padding for FFT z Zero-padding Quiz! y. Purpose x. N’=N+a=2^n, easier for FFT computation x. Interpolation to have better resolution y. Example xfft. Zero. Padding 01. m -29 -

Example: Down-sampling z Down sampling y. High-frequency components are missing yfft. Resample 01. m -30 -

Harmonics z. Why do we have harmonic structures in a power spectrum? y. Since the original frame is quasi-periodic… z. If we want to use the spectrum for pitch tracking, we need to enhance the harmonics. How? -31 -

How to Enhance Harmonics? z. Goal y. Enhance harmonics for pitch tracking z. Approach y. Take an integer number of fundamental periods Hard! y. Use windowing y. Use zero-padding for better resolution -32 -

Example: Harmonics Enhancement (1) zfft. Harmonics 01. m -33 -

Example: Harmonics Enhancement (2) zfft. Harmonics 02. m -34 -

Example: Harmonics Enhancement (3) zfft. Harmonics 03. m -35 -