1 Discrete-Time Fourier Transform (DTFT)

1 Indian Institute of Technology BombayDept of Electrical EngineeringHandout 11EE 603 Digital Signal Processing and ApplicationsLecture Notes 4 September 2, 20161 Discrete-Time Fourier Transform (DTFT)We have seen some advantages of sampling in the last section. We showed that by choosingthe sampling rate wisely, the samples will contain almost all the information about theoriginal continuous time signal. It is very convenient to store and manipulate the samplesin devices like a times the samples need to be processed beforeplaying it back(reconstructed).A branch of signal processing known as Digital Signal Processing (DSP) deals entirely withthis.

2 This should be contrasted with the naive way of reconstructing the continuous timewaveform first and then applying analog processing techniques. Let us enumerate the Do all the processing/computations on the samplesx[n],n Zand perform afinal reconstruction to obtain a signaly(t).2. Given the samplesx[n],n Z, first reconstruct the signalx(t), and then doanalog processing to generatey(t).An important question is whether both the approaches give the samey(t)from a given setof samples. If the answer is affirmative, the first method will be a clear winner in termsof feasibility and convenience, since we have the freedom of using devices like computer,Digital Signal Processors (DSPs).

3 The primary data processing application we have in mindis filtering, which is nothing but frequency shaping of the input signals. Linear filteringinvolves convolution by the impulse response of the filter. Equivalently we have to multiplythe responses in the frequency domain. What is the frequency domain representation of asampled Do samples have frequencySamples, after all, are a given set of values. How can they have a frequency associatedwith it? To find this, it is fruitful to look from the reverse angle. Given a set of samples,what could have been the original waveform from which these samples were taken. Thiswill automatically identify the samples with some time intervalT.

4 For each value ofTyou may find a possibly differentparentwaveform. Furthermore, for a givenTthere canby multiple parent waveforms which generate the samples. All these connections implythat the frequency components that we attach to the samples have to relate to all possibleparent waveforms. Thus it makes sense to fixT, as a real number such thatx[n]=x(nT)wherex[n],n Zare the samples andx(t)the corresponding parent waveform, which hasa continuous-valued index. Suppose we know a signalx(t)whose samples are the givenpointsx[n],n Z. There is an intimate relationship between the Fourier TransformX(f)ofx(t)and the frequency contents ofx[n].

5 In fact our generalized Fourier Transform allowsus to view both the continuous and discrete signal as part of the same tapestry. Here ishow it is done, illustrated by an the following signal which is a segment from the output of a (t)tMany a times we cannot afford the luxury of waiting till the end of the signal andthen computing the FT. So we wait for a reasonable delay, crop the signal (multiply bya window) and take its Fourier Transform . The dashed lines shows the points where thesignal is windowed (segmented). Consider the first segment, call itx(t).s(t)rect (t)tThe FTX(f)ofx(t)is not bounded in time (why?). Assume thatX(f)is given bythe following figure (This is not an exact computation, only a visual aid for pedagogicalpurposes).

6 One more comment is in order here. In most plots of the FT, we will simplyshow the magnitude alone. The reader is expected to imagine some phase componentexp(j (f)), where (f)is an odd function if the signalx(t)is real, (f)= ( f).We will deal with the phase component in a more precise manner later, when we designpractical (f)123 1 2 3It is very difficult to keep track of all the frequencies in[0, ]. A practical approach is tolimit the components to the significant values. For example, we consider the values outside[ fm,fm]to be insignificant for all computational purposes. This is not only reasonable,but also of great help in analytical (t)tFT fX(f)Once the frequency components are limited, we know that the signal can be loss-less-ly stored by 2fmsamples per second, wherefmis the highest frequency content is illustrated in the following figure, where sampling and the corresponding FourierTransform are shown1, whereT= (nT)tf X(f)Figure 1: FT of the time samplesThe FT of the samples is given by an appropriate repetition ofX(f)by the convolution-multiplication theorem.

7 We can sample at any rate greater than 2fmsamples per second,In particular, X(f)=SRQnx(nT) (t nT)e j2 ftdt(1)=Qnx(nT)exp( j2 fnT).(2)We have already introduced the notation thatx(nT)=x[n]. Using this X(f)=Qnx[n]exp( j2 fnT)(3)1In the figures, we will use the symbolto denote that there is a repetition of the pattern in thedirection of the symbol, see for example Figure that the RHS of (3) is 2fm-periodic inf. This also tells that the samples ofx(t)are taken at integer multiples ofT=12fm. However, once the samples are taken, they aredead values, in the sense that they can be written on a piece of paper, or stored in memoryarrays in diverse fashion.

8 It thus makes sense to define a Fourier Transform for our sampleswhich is somewhat independent of the sampling interval, but only depends on the sequenceof sample-values. This can be achieved by a suitable scaling. In particular, X(f) = X(fT)=Qnx[n]exp( j2 fn).(4)Note that X(f)has unit period, we call this the DTFT ofx[n]. From our generalizedFourier Theory, the inverse of DTFT should correspond to the input samples, which arespaced at unit intervals. However, we have learned that for a periodic waveform, thegeneralized Fourier representation is obtained by computing the Fourier Series ,x[n]=S12 12 X(f)e+j2 fndf.(5)Notice the slight difference from the original FS formula.

9 Here we usee+j2 ft, to be con-sistent with the formula for DTFT in (4).DTFT and Inverse-DTFT X(f)=Qnx[n]exp( j2 fn)(6)x[n]=S+12 12 X(f)exp(+j2 fn)df(7)2 Digitizing Signals and SystemsImagine a bandlimited waveformx(t), with the frequency domain representationX(f), Transform ofx(t). Sincex(t)is bandlimited, it is unlimited time . Ifx(t)is real,X(f)is complex symmetric around the origin. However, there is no reason that our signalshould be real, and we will consider complex signals of the form,x(t)=xR(t)+jxI(t).For illustration, we will consider ax(t)waveform with Fourier TransformX(f)as (t)FT fX(f) 2+ 2We will repeatedly use the above pictorial representation.

10 We should keep couple of thingsin mind while using these pictures. First, the inverse Fourier Transform (IFT) of suchanX(f)may correspond to a non-causal signal. Thus our descriptions are not for theabsolute time , and we do not really worry about causality. Secondly, we do not emphasizethe phase information in these pictures. This is partly due to the fact that we expect thephase response to be linear in the frequencies of interest, this will be covered Frequency Response of a SystemThe frequency response of a LTI system can be determined by sending adequately spacedcomplex sinusoids and measuring the output attenuation, in terms of magnitude scalingsand phase-rotations.