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**Unformatted text preview: **CTFT Continuous-Time Fourier Transform no): I f(t) e "1"" dt 7 -00
[continuous, non-periodic] (r 00 f(t)=(1[2_n) I F(o)eimdo ' [continuous, non-periodic] 'T. Discrete-Time Fourier Transform 'X(ej“’ = Z--’v>'<(n)e‘j""J _- m
nap \ [continuous,_periodic] _ Ix(n)= (1127:) i X(ej‘°)ej“°° dw [discrete, non—periodic] FOURIER SERIES P12 ' t " C(k) = (HP) fa) e -i2:rktIP dt I -P1’2 ' [discrete, non-periodic]
i? r . I..- k: __°c_, - [continuous, periodic] FT 2: FFT' Discrete FouriergTransform
(fast version is Fast FOurier-Transform FFT) 7-‘- N-‘l I? = Z xm) e-j(21dN)nk n'=0 [d iscrete,_ periodic] N-“l I , X01) =(1/N) I; X(k)ei(2.m’N)nk [discrete, periodic] Notes on Fourien Map (1) There are four Fourier-transform-like pairs here that describe the same functions
in the time domain (t or n) and in the frequency domain (0,0), or k) (2) One of the pairs (CTFT) is an integral/integral pair, one (DFT) is a
summation/summation pair, and the remaining two involve an integrallsummation
pairing. (3) Both time and frequency descriptions may be either continuous or discrete (hence the
four possible cases. To see if the description is continuous or discrete, consider ﬁrst
whether the equation is an integral (continuous) or a summation (discrete). Also, the
frequency variable for continuous time is t, while that for discrete time is n. For
frequency, discrete frequency is k, while continuous frequency is Q or (u. (4) The continuous frequency variable Q is considered to be in physical units of radianslsec.
Thus frequency in Hertz would be 0121:. The continuous frequency to has various
interpretations. One is that it is rad ians/second where the sampling frequency is set to
1 Hz (21: radian/second, a normalized sampling time T=1). In this view, the sampling
frequency in Hertz is f5, the sampling time is T=1/fs, and w = QT. We could rewrite the
equations for the DTFT using QT for (.0. Possibly the most useful interpretation of u) is
that it is a dimensionless angle. it is the angle in the unit circle in the z-plane.
Sometimes this is called radians/sample. (5) Note that for each pair, when the description in one domain is periodic, the description
in the other domain is discrete (think Fourier Series). When the description in one
domain is non-periodic, the description on the other domain is continuous. (6) The Fourier Series and the DTFT are “cross” cases where [discrete, non-periodic]
is paired with to [continuOus, periodic]. Mathematically, they are the same except the
roles of time and frequency are reversed (plus a few minor notational conventions). (7) The DTFT is thus a Fourier Series for a periodic function of frequency and the “Fourier
Series coefficients” are the discrete samples in time. Thus the DTFT is always
periodic (with a period, the spacing of the sampling replicas) equal to the sampling
frequency (which is 27: for the normalized case on the map). This periodicity in
frequency is the direct consequence of sampling in time. (8) For the DTFT, we might think of x(n) as a "signal or it might be the “impulse response”
- of a filter, which we would probably denote h(n). The DTFT of a signal is usually called
its “Spectrum” while the DTFT of an impulse response is called the “Frequency
Response.” Mathematically there is no difference, but use the terminology right: a
signal has a spectrum while a filter has a frequency response. if at some point you
become confused about the nature of the DTFT, it is often useful to think of it in terms
of the nature of a frequency response (a prime example of a DTFT). (9) The top two boxes relate to continuous time, while the bottom two relate to discrete
time. Thus time sampling occurs in the bottom boxes. The boxes on the left relate
to continuous frequency (like turning the knob on a function generator) while the
boxes on the right relate to sampling in frequency. The DFT is a sampling (samples at frequencies 21clN) of the DTFT. (1 0) As noted the bottom/left to top/rig ht pairs are "Fourier Series” cases. Looking now
instead at the top/left and bottom/right pairs we see that the DFT maybe thought of as
a sort of numerical integration of the CTFT. However, this numerical integration means that both time and frequency have to be sampled, and hence are periodic, so
this complicates our thinking. (11) Where is the famous “Laplace transform?” The LT is the father of all the transforms
here. The LT relates to the entire “s-Plane” while the Fourier transforms here relate only to the imaginary axis in the s—plane (“real” or physical frequencies). (12) And where is the "z-Transform?” This makes the LT the grandfather of the DTFT and
the DFT. The z-transform is the LT evaluated for a sampled time function, and the z-
transform applies to the entire "z-plane.” The DTFT applies only to the unit circle in the
z—plane and the DFT is further restricted to only discrete points on that unit circle. (13) And what about the very famous Fast Fourier Transform (F FT)? The FFT is a fast
algorithm for computing the DFT. The FFT (as an algorithm) is not the DFT (a brute-
force summation). But the F FT of a sequence x(n) is identical to the DFT of the same
sequence x(n). It is not an approximation. Because the two results are the same, the
terms are often used interchangeably. It perhaps makes sense to use FFT rather then
DFT because (a) of the danger of confusing the terminology DFT withDTFT, and (b)
because virtually all DFT’s are actually computed by FFT. in fact, virtually all the
transforms in the four boxes are (or can be, or for practical purposes must be) computed using the FFT. conventional DT'F‘T',rwe would need to normalize the frequency axis so that the sampling
frequency is 21: (without any physical dimensions), usually by thinking of T as having a
dimensionless value of '1 ' ' 2b“ 12iRECTLWEQIEIEQIBED;_'T'FT‘ The Continuous "Time Fourier ‘T'ra'nsform (CTFT), X49), of a signal xa(t) is given by: 00 x352): f xén) e'jmdt ‘ I [CTFT] (38) -cc c??? 0?? . .
Aux-egos {Inverse (3h) “with inverse:
w
* n .
‘ xa(t) = (1121:) f Xa(Q)e’ ‘do. ~90 while the usual form of the Discrete Time Fourier Transform (DTFT). Helm), of a discrete time
sequence x(n) is: - ' ' 00 X(ejf°) = E x(n) em n=---oo [Usual DTFT] (4a) with inverse: 1;
'x(n) = (1127:) X(eJ'°3) einw d0) [Usual inverse o‘rm (4b) I '11; Here the frequency co is the "normalize " version of frequency, which is considered to be
' dimensionless. This can be related to the physical frequencies (2 (radians/second) orf (Hz, or
cycleslsecond) as: a) = 2::(Qlﬂs)= 21:62:35): 2m = or H (5) where 93 and is are the corresponding sampling frequencies, and Teri/fs is the sampling period. SinCe the CTFT is expressed in terms of physical frequency, it will be very usefui to also
represent the DTFF‘ and its inverse interms of 52 instead of co That is, we insert QT for co, and
this results in the equations: ' A BEODSP: SAMPLING (5) EN'#200 (9) - -co . mtg.) X(el',QT’)E—-_ 2 ‘x(n)e;5”9T .. , ‘ 1" Physical -(6la). 1‘ n=.'oo
. a: =‘- SM
‘5 _ x(n) = (1721:) xw'n'rjeinmidh' :. {P.hys’iCal Inverse DT'FT] - ’{éb} c , I ‘ y where the limits on the integral from (927-2: to 7: become to «IT in terms of (0421', and the "f' F outside the integral results from day = T69, ail the rest of the reformation being simple substitution. ' At this point we want to look. at the case xin) = xa(t=n"i‘) where 'T' is the sampling intei'vai. “This
precedute (Which we ﬁnd in 1975 era DSP textbooks) will lead us to the same teetiit of spectra!
replication that we ﬁnd with "impulse sampling We begin with equation 3b , the inverse .CTFT', : . which is true for at! t, and therefore certainty for"t=nT ' - 00 xa(t"='h‘r) = (1121:) xgkzjéiﬂnT dh (7-8} -CD The integral cvei: inﬁnite ii its in eq ‘tipn (SM-can be represented as at infinite sum over ﬁnite '
i segments of length 2an ' ‘ "- _ w n={2m}+-1)«dt' ' ., _
xa(t‘—.‘n'I} = (mu) 2 Xam) eiQnT‘ an (Tb) m=m mam-‘1):le By e change'of variable, Q 7—:"9‘ -2rn§d‘T", We have; ' oo barf!" I stein ='. we 53 j" $495 ei‘t'iT' 6.51) ' ' ﬁe)"
i ‘ than Q'=--7'cfi' w ‘ '
or (since (19' = (in):
on 1:11” 7 ‘ ' I
we"). .= £11219 2 .i" Xaiﬁ‘t 2W) amt-1W3"? .ds'z 7 ' ...(7d>
“IF-<50 1 --'7dT b
' ‘ I . " : :I
x - EN#200 (10) i ' BEODSP - SAM PLINQ (6)
a I .‘ Recognizing that e W=1 for an integers m and n, and multiplying by TIT '=1 we have: no MT
_ xa(t=n'T} = (mu) 23 (111) xam +21rmﬂ) e19?“ d9 (Te)
m=4oo N Reven‘sing the tides" of sumrﬁ'aﬁbh aim} integratiqh: ‘ physical inverse DT'FT. with: x9507) = mm 2 xa(sz+-2xmm}' 1 ‘ (8a) mice Noting that 21W = 2m ; (23, the sampling frequency in radiansfsecond: 7 h / ‘ (8b) which bleryyig, slspﬁe‘ reboatiohs, |§paéed at int-await; bf this 5331;51an fr’éqqency, anq scaled by1f‘r' : In tgrn'gs pf‘coI" , ' ’ . ' .w’ mam} = mm 2 Homes): ‘ I, m . mam) = :(1rr) E- xa(mrf-+2nmn) ‘ (9a) =00
or. 9° x6429): ; I km) 2 Kano) fgwmﬂl ' ' ‘” . (913$ m=o<2
showing séeétfal lrepli‘cas at spacing 2n ' : ‘ BEoDSP - SAMPLING (7) EN#200 (11)
3‘24 ' u .‘ ‘ ' - r ‘ , ' Elma. Dean. 7 U I <_—=7
Fr. .-
a ti? ° 4529—9
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