What is the frequency domain representation?
X{e^jwn}, w: normalized frequency in radians
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.1 Review of CTFT Definition
Xidian University
jimleung@
Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain Outline Review of continuous-time Fourier transform (CTFT) Discrete-time Fourier transform (DTFT) DTFT theorems DTFT computation using MATLAB
X_k does not convergence to X for all the values of frequency, but convergences in the mean-square sense.
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Digital Signal Processing
© 2013 Jimin Liang
Omega: radians/sec Polar form
(1) Magnitude spectrum: |X_a|
(2) Phase spectrum: arg{X_a}
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.1 Review of CTFT Dirichlet conditions (1) finite discontinuities, finite number of maxima and minima in any finite interval (2) absolutely integrable
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain Objective of this lecture Time domain representation of a DT signal x[n] = sum_k(a_n delta[n-k])
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Convergence condition Absolutely summable is a sufficient condition. Example 2.9
xn 0.5 n n
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Convergence condition Definition: partial sum (1) Absolutely summable (uniform convergence)
Amplitude
0.6 0.4 0.2 0 -0.2 0 0.2 0.4 0.6 0.8 1
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Convergence condition (3) Dirac delta function: for sequences that are neither absolutely summable nor square-summable.
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Symmetry relations Table 3.1, 3.2 Example 3.7
at a prescribed set of discrete frequency points
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Digital Signal Procete-Time Signals in Frequency Domain 3.6 DTFT computation using MATLAB
(1) Parseval’s theorem
(2) Energy density spectrum
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.1 Review of CTFT Band-limited continuous-time signals (1) Full-band vs. band-limited
Real part 2 1 0.5
© 2013 Jimin Liang
Digital Signal Processing
Chapter 03-1-Discretre-Time Signals in Frequency Domain
Dr. Jimin Liang School of Life Sciences and Technology
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.3 DTFT theorems Modulation: 时域相乘->频域卷积
Parseval’s theorem
3.4 Energy density spectrum
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.6 DTFT computation using MATLAB MATLAB functions: freqz, abs, angle, real, imag, unwrap The function freqz can be used to compute the values of the DTFT of a sequence, described as a rational function in the form of
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Example 3.8
1 K=5 1 0.8 K=10
(1) Independent of K, there are ripples around w_c. (2) K increases, the number of ripples increases, but the height of largest ripple remains the same. (3) K->inf, error->0 Gibbs phenomenon
X_k convergences to X for all the values of frequency.
If a sequence is absolutely summable, its DTFT exists.
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Digital Signal Processing
© 2013 Jimin Liang
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Convergence condition (2) Mean-square summable:
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.3 DTFT theorems Convolution: 时域卷积->频域相乘
Proof:
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Digital Signal Processing
Synthesis equation:
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Signals in Frequency Domain 3.2 Discrete-time Fourier transform (DTFT) Basic properties of DTFT Different forms of expression
0.8
Amplitude
0.6 0.4 0.2 0 -0.2 0 0.2 0.4 0.6 0.8 1