2.3. 1 Discrete Fourier Transform DFT There are four cases for the fourier transform of an infinite-length continuous signal(Fig 263) mmmcaeT 2-20T/2 2072J。了 x () 灬A Fig 2.63 Types of Fourier transform
There are four cases for the Fourier transform of an infinite-length continuous signal (Fig. 2.63). 2.3.1 Discrete Fourier Transform (DFT) Fig. 2.63 Types of Fourier transform
2.3. 1 Discrete Fourier Transform DFT) uFig 2.63(a): a nonperiodic continuous signal x(t)and its Fourier transform spectrum X(. The spectrum is continuous OFig. 2.63 (b): a periodic continuous signal and the frequency spectrum is or discrete F:X()=「x(012dt (2201) JFT:x()=∑X()e2m (2202) where f=ky(k=0,±1±2…) Af: fundamental frequencys
❑Fig. 2.63 (a): a nonperiodic continuous signal x(t) and its Fourier transform spectrum X(f). The spectrum is continuous. ❑Fig. 2.63 (b): a periodic continuous signal, and the frequency spectrum is or discrete. where Δf: fundamental frequency, 2.3. 1 Discrete Fourier Transform (DFT) ( ) ( ) − − = 2 2 1 2 : T T j f kt k x t e dt T FT X f (2.201) ( ) ( ) =− = k j f t k k IFT x t X f e 2 : (2.202) f = kf (k = 0,1,2, ) k T f 1 =
2. 3. 1 Discrete Fourier transform DFT uFig 2.63(c): the Fourier transform of a nonperiodic discrete signal The Fourier transform of an infinite-length discrete time sequence is a periodic continuous spectrum FT:X()=∑ x(t, e2m (2203) IFT:x(t, 乙X(/2d 2.204) Whee〃sn△(n=0,±1+2…) At is the sampling period; fs is the sampling frequency of the time sequence
❑Fig. 2.63 (c): the Fourier transform of a nonperiodic discrete signal. The Fourier transform of an infinite-length discrete time sequence is a periodic continuous spectrum. where Δt is the sampling period; fs is the sampling frequency of the time sequence. 2.3.1 Discrete Fourier Transform (DFT) ( ) ( ) =− − = n j f t n n FT X f x t e 2 : (2.203) ( ) ( ) − = 2 2 1 2 : s s n f f j ft s n X f e df f IFT x t (2.204) t = nt(n = 0,1,2, ) n s f t 1 =
2.3. 1 Discrete Fourier Transform DFT uFig 2.63(d): the Fourier transform of a periodic discrete time sequence. Its spectrum is also periodic and discrete The sampling period is At, then T=N△t
❑Fig. 2.63 (d): the Fourier transform of a periodic discrete time sequence. Its spectrum is also periodic and discrete. The sampling period is Δt, then 2.3.1 Discrete Fourier Transform (DFT) T = Nt
2. 3. 1 Discrete Fourier Transform DFT) Conclusion: For a periodic x(t the spectrum X( is bound to be discrete and vice versa ☆lfx() is nonperiodic,then×(is continuous, and vice versa the case shown in Fig. 2. 64 (d)where both the time and frequency signals are discrete and periodic provides us with the possibility of using a computer to implement spectrum analysis
Conclusion: ❖For a periodic x(t) the spectrum X(f) is bound to be discrete, and vice versa. ❖If x(t) is nonperiodic, then X(f) is continuous, and vice versa. ❖The case shown in Fig. 2.64 (d) where both the time and frequency signals are discrete and periodic provides us with the possibility of using a computer to implement spectrum analysis. 2.3.1 Discrete Fourier Transform (DFT)