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EE-240/2009 Análise Espectral. { x ( nT ), n  Z } = sinal x ( t ) seja amostrado com período de amostragem T . D iscrete- T ime F ourier T ransform:. Transformada de Fourier . { x n = x(nT) } registrada somente para n = 0, ..., N – 1 .

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slide1

EE-240/2009

Análise Espectral

transformada de fourier
{ x(nT), n Z } = sinal x(t) seja amostrado com período de amostragem T.

Discrete-Time Fourier Transform:

Transformada de Fourier
  • { xn = x(nT) }registrada somente para n = 0, ..., N – 1.
  • DTFT discretizada em N pontos entre w= 0 a w = 2p.
  • Discrete Fourier Transform:
slide6
randn('state',0)

x = randn(5,1);

N = length(x);

X = zeros(N,1);

for k = 0 : N - 1

n = 0 : N - 1;

Wk = exp(- j * 2*pi *k * n / N)' ;

X(k+1) = sum(x .* Wk);

end

[X fft(x)]

absX = abs(X);

[sum(x.^2) sum(absX.^2)/N]

Energia do sinal =

Teorema de Parseval

slide9

N par

N ímpar

slide10

fft(x)

2

3

1

4

k = 0

N = 8

7

5

6

an lise conjunta tempo freq ncia1

Segmento 5

Segmento 2

Segmento 3

Segmento 4

Segmento 1

DFT

DFT

DFT

DFT

DFT

Análise Conjunta Tempo-Freqüência
slide20

B = SPECGRAM(A,NFFT,Fs,WINDOW,NOVERLAP) calculates the spectrogram for the signal in vector A. SPECGRAM splits the signal into overlapping segments, windows each with the WINDOW vector and forms the columns of B with their zero-padded, length NFFT discrete Fourier transforms. Thus each column of B contains an estimate of the short-term, time-localized frequency content of the signal A. Time increases linearly across the columns of B, from left to right. Frequency increases linearly down the rows, starting at 0. If you specify a scalar for WINDOW, SPECGRAM uses a Hanning window of that length. WINDOW must have length smaller than or equal to NFFT and greater than NOVERLAP. NOVERLAP is the number of samples the sections of A overlap. Fs is the sampling frequency.

exemplo
CHIRP Swept-frequency cosine generator.

Y = CHIRP(T,F0,T1,F1) generates samples of a linear swept-frequency signal at the time instances defined in array T. The instantaneous frequency at time 0 is F0 Hertz. The instantaneous frequency F1 is achieved at time T1. By default, F0=0, T1=1, and F1=100.

Exemplo
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t=0:0.001:2; % 2 secs @ 1kHz sample rate

% Start @ DC, cross 150Hz at t=1sec

x = chirp(t,0,1,150);

specgram(x,256,1000,256,250);

transformada wavelet
Transformada Wavelet

Idéia: Janelas de tamanho variável

Janelas estreitas componentes

localizados no tempo

(picos estreitos)

"Janelas" largas para

baixa freqüência

(variações de linha de base)

transformada wavelet1
Transformada Wavelet

Deslocamento

“Onda localizada (b)”:

Média zero e duração limitada (a).

Wavelet Mãe

Escala

slide29

CWT Real or Complex Continuous 1-D wavelet coefficients.

COEFS = CWT(S,SCALES,'wname') computes the continuous wavelet coefficients of the vector S at real, positive SCALES, using wavelet whose name is 'wname'. The signal S is real, the wavelet can be real or complex.

COEFS = CWT(S,SCALES,'wname','plot') computes

and, in addition, plots the continuous wavelet

transform coefficients.

slide30

t = 0:0.001:2; % 2 secs @ 1kHz sample rate

% Start @ DC, cross 150Hz at t=1sec

x = chirp(t,0,1,150);

cwt(x,1:64,’gaus1’,’plot’);