电子科大数字信号处理课件

- () called the principal value.
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The DTFTs of some sequences exhibit discontinuities of 2 in their phase responses. An alternate type of phase function that is a continuous function of is often used. It is derived from the original phase function by removing the discontinuities of 2.
X a ( j) X a ( j) e ja ( )
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3.1.2 Energy Density Spectrum
The total energy x of a finite-energy continuous-time complex signal xa (t ) is given by: 2 x xa (t ) dt
Time-domain k a k [n k ] (a weighted linear combination of delayed unit sample sequences.)
Transform-domain a. frequency domain b. Z domain (a sequence in terms of complex exponential sequences of the form{ e jn } and { z n }.

Applying the CTFT to both sides, we have:
Ya ( j) H a ( j) X a ( j)
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3.2 The Discrete-Time Fourier Transform
It’s an useful tool to represent a discretetime signal in frequency-domain. It briefly called DTFT. The relationship between it’s time and frequency is: discrete and non-periodic in time-domain, continuous and periodic in frequency-domain.
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Several Forms of FT
FT-continuous in time,continuous in frequency
X ( j) x(t )e jt dt

1 x(t ) 2
x(t)


x( j)e jt d
X ( j )
t

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Result:
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The DTFT X(ej) of a sequence x[n] is a continuous function of It is also a periodic function of with a period 2:
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3.2.2 Basic Properties
In general, X(ej) is a complex function of the real variable and can be written as:
X(ej) = Xre(ej) + j Xim(ej) Where,Xre(ej) and Xim(ej) are, respectively, the real and imaginary parts of X(ej) , and are real functions of . X(ej) can alternately be expressed as: X(ej) = | X(ej) |ej() where () = arg{X(ej) } 17
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So: X(ejω)is a periodic function of , and the DTFT definition equation is the Fourier series expansion of the periodic function X(ejω) , x[n] are the coefficients of the Fourier series.
Continuous function in time domain will cause non-periodicity in frequency domain,whereas the non-periodicity in time domain will cause Continuous function in frequency domain.
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3.1.1 The Definition
The definition of CTFT is:
X a ( j) xa (t )e


jt
dt
The CTFT often is referred to as the Fourier spectrum. The I-CTFT(inverse CTFT) is:
X ( jk 0 )
2 0 T0
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T0
Result:
Continuous function in time domain will cause non-periodicity in frequency domain,whereas the periodicity in time domain will cause discrete frequency spectrum.
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3.1.4 The Frequency Response of an LTI Continuous-Time System
The output response ya (t ) for an initially relaxed linear, time-invariant continuoustime system characterized by an impulse response ha (t ) for an input signal xa (t ) is given by : ya (t ) ha (t ) xa ( )d
An ideal band-limited signal has a spectrum as:
0, 0 a , X a ( j) 0, b .
In practice, for a band-limited signal outside the specified frequency range, the signal energy is arbitrarily small. The bandwidth of a band-limited signal. Lowpass, highpass, bandpass.
| X(ej) | is called the magnitude function. () is called the phase function. Both quantities are real functions of . In many applications, the DTFT is called the Fourier spectrum. Likewise, | X(ej) | and () are called the magnitude and phase spectra. For a real sequence x[n], | X(ej) | and Xre(ej) are even functions of , whereas, () and Xim(ej) are odd functions of .
1 xa (t ) 2



X a ( j)e jt d
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We denote the CTFT pair of above two equations as: CTFT xa (t ) X a ( j)
The CTFT is a complex function of in the range . It can be expressed in polar form as
Chapter 3 Discrete-Time Fourier Transform
The Definition The Theorems The FrLTI DTS Phase and Group Delays
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How to Represent the DiscreteTime Signal?
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3.2.1 Definition
The discrete-time Fourier transform of a sequence x[n] is defined by X (e j ) :
X (e )
j
n
x[n]e

j n
It’s inverse transform process: 1 j j n x[n] X (e )e d 2
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The convergence condition of above series is:
n


x[n]e
j n

n


x[ n]
Result:Discrete time will cause periodicity in frequency,and the non-periodicity in time domain will cause continuous function in frequency domain.
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FS-continuous in time,discrete in frequency