C Williams W Alexander NCSU THE DISCRETE FOURIER TRANSFORM ECE 513 Fall 2019

C williams w alexander ncsu the discrete fourier

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C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 139 / 193
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The DFT for Spectral Estimation We note several things POINT 1: Even with large L , the spectrum of a rectangular window have very large sidelobes This causes a distortion that cannot easily be eliminated by increasing L POINT 2: Signal are often of some set duration L In most practical applications, we cannot easily and arbitrarily increase L C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 140 / 193
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The DFT for Spectral Estimation To address these problem, other window functions, w ( n ) can be used These windows have slightly different spectral characteristics that the spectrum of the rectangular window An example of such a window is a Hanning window w ( n ) = ( 1 2 ( 1 - cos 2 π L - 1 n ) , 0 n L - 1 0 , otherwise (93) The sidelobes of the Hanning window are significantly smaller that those of the rectangular window Its main lobe, however, is approximately twice as wide C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 141 / 193
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Resolving Close Frequencies The size of the main lobe of the window function determines whether or not two adjacent frequencies can be resolved Consider a signal that is a sum of two sinusoids x ( n ) = cos( ω 0 n ) + cos( ω 1 n ) (94) For a rectangular window of length L , this results in the following spectrum ˆ X ( ω ) = π [ W ( ω - ω 0 )+ W ( ω - ω 1 )+ W ( ω + ω 0 )+ W ( ω + ω 1 )] (95) C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 142 / 193
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Resolving Close Frequencies Distortion can cause us not to be able to distinguish one spectral line from the other This occurs if | ω 0 - ω 1 | < 2 π L If | ω 0 - ω 1 | < 2 π L , the two window functions W ( ω - ω 0 ) and W ( ω - ω 1 ) overlap As a result, the two spectral lines of x ( n ) become indistinguishable If | ω 0 - ω 1 | ≥ 2 π L , we will see the two distinct peaks corresponding to the individual spectral lines NOTE: This is the case for a rectangular window. This becomes worse if the main lobe is wider, as is the case for the Hanning window C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 143 / 193
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Resolving Close Frequencies Example (7.8) Consider the following signal x ( n ) = cos( ω 0 n ) + cos( ω 1 n ) + cos( ω 2 n ) (96) where ω 0 = 0 . 2 π , ω 1 = 0 . 22 π , and ω 2 = 0 . 6 π Let N = 2 15 . Plot the DFT for L = 25 , 50 , and 100 For the sampling frequency F s = 1 kHz , these frequencies correspond to F 0 = 100 Hz , F 1 = 110 Hz , and F 2 = 300 Hz C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 144 / 193
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Resolving Close Frequencies Example (7.8) The figure below show the resulting spectrum for N = 2 15 , L = 25 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 F (Hz) |X(k)| Spectrum of Finite Duration Signal of x(n), N = 2 15 , L = 25 Figure: Spectrum of Finite Duration of x ( n ) . C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 145 / 193
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Resolving Close Frequencies Example (7.8) The figure below show the resulting spectrum for N = 2 15 , L = 50 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 30 35 F (Hz) |X(k)| Spectrum of Finite Duration Signal of x(n), N = 2 15 , L = 50 Figure: Spectrum of Finite Duration of x ( n ) .
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