Determine the solution of the difference equation? Differentiate DTFT and DFT.

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Babita Majumder · Answer

The DTFT(Discrete Time Fourier Transform) is nothing but a fancy name for the Fourier transform of a discrete sequence. It is defined as:

The frequency variable is continuous, but since the signal itself is defined at discrete instants, the resulting Fourier transform is also defined at discrete instants of time. The number of time points will still be infinite.Its just that between any two time points you would have a finite number of points.We also know that the Fourier transform of a sampled signal is a series of replications of the spectrum of the original signal at frequencies spaced by the sampling frequency.

Mathematically, this is expressed by

Here T is the sampling period.So this is an alternate mathematical expression for the DTFT.As we can see, the DTFT is periodic, with period equal to the sampling frequency. Hence we normally represent DTFT over a single period as shown below. This is implicit in the definition.

Mathematically this periodicity can also be seen by noting the periodicity of the discrete(in time) exponential

2.Discrete Fourier TransformBut the DTFT is difficult to evaluate on a computer, since a computer works only on finitenumber of points. So to make the evaluation of the DTFT possible on a computer, we choose a finite number of frequency points. This is equivalent to sampling the Fourier transform at a certain number of points.This is called the Discrete Fourier Transform(DFT).The general convention is to use N frequencies separated by 2*pi/N radians.2*pi corresponds to the angular frequency variation over one cycle of the original waveform(the cycle over which we take the N time samples) . We divide this into N frequency 'bins' which we index by an index variable k.

Mathematically, this is expressed as

Since the DFT is basically the sampled version of the DTFT, it is also periodic, with periodN(the number of frequency samples taken). Hence the DFT is also represented over one 'period'(discrete frequency period - N). This is implicit in the definition.Mathematically, this periodicity can be see by noting the periodicity of the discrete(both in time and frequency) exponential

3.Discrete Fourier SeriesThe Fourier Series uses an infinite number of complex exponentials to represent a signal. Even though the signal frequencies are discrete, there are an infinity of them. The number of frequency points will be infinte. However between any two frequency points you'll have a finite number of frequency points. So you have a discrete spectra.To make the evaluation of a Fourier series possible on a computer, we choose a finitenumber of frequency points(complex exponentials).This is called a Discrete Fourier Series(DFS).(Since the Fourier series can be used only for periodic signals, you have a finite number of samples over a single period of the signal).Again as with the Discrete Fourier Transform, the convention is to use N frequencies separated by2*pi/N radians. Here2*pi corresponds to the angular variation over one period.We divide this into N frequency 'bins' which we index by an index variable k.A discrete signal is expressed using the DFS as follows:

where the complex coefficients are given by:

As you can see, this is a linear combination of N discrete complex exponentials:

The DFS is also periodic with period N(the number of samples taken over one period). Hence it is also represented over one period.Mathematically, this periodicity can be seen by noting the periodicity of the complex coeffecients:

In which cases is each one used? The Fourier series and the Fourier transform can both be used for periodic and aperiodic signals.A periodic signal can be expressed in the time domain as a Fourier series, which is nothing but a series of exponentials. Now we know that the Fourier Transform of an exponential function is an impulse. So if we take the Fourier transform of this signal, we have a series of impulse functions.The Fourier transform of a periodic signal is thus a series of impulse functions at the harmonic frequencies. It is a discrete spectra, just like what we obtained by the Fourier series. There is no new information in this. Its just stating the fact that 'The Fourier transform of a periodic signal is discrete".Similarly we can represent an aperiodic signal using Fourier series as follows.We can create a periodic function by summing up(repeating) an infinite number of instances of an aperiodic function:

where P is the period of the resulting periodic function.Now, fPfP can be expressed as a complex Fourier series and it can be shown that the Fourier series coefficients are proportional to the samples of the Fourier transform of fftaken at intervals of 1/P. This is actually a particular case of the Poisson Sum Formula.See here for a proof(Page - - Section -8.2.1):http://www.siam.org/books/ot/...However remember that there are convergence issues. So one of the two might be more suitable in some cases. (Will write more about this when time permits).Now, to evaluate either Fourier Transform or Fourier Series on a computer, you need to make the number of points in both time and frequency domains finite. Time Domain Discretization:For a periodic signal, since the information over one period describes the signal for all time, you just need to take a finite number of time samples over one period and you are done.For an aperiodic signal, you need to time limit the signal and take samples over this period to discretize it in the time domain.Frequency Domain Discretization:For frequency domain discretization, you just take evenly spaced points over a complex circle(2*pi radians) - i.e one 'period'(for an aperiodic signal, period here represents the interval over which it has been time limited.) Updated 3 Aug • View Upvotes

Anas Kallara Dirtyss 4k Views n mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data (samples) whose interval often has units of time. From only the samples, it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by thesampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the Discrete Fourier Transform (DFT) (see Sampling the DTFT), which is by far the most common method of modern Fourier analysis.Both transforms are invertible. The inverse DTFT is the original sampled data sequence. The inverse DFT is a periodic summation of the original sequence. The Fast Fourier Transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.The utility of this frequency domain function is rooted in the Poisson summation formula. Let X(f) be the Fourier transform of any function, x(t), whose samples at some interval, T (seconds), are equal (or proportional to) the x[n] sequence, i.e.

. Then the periodic function represented by the Fourier series is a periodic summation of X(f). In terms of frequency

in hertz (cycles/sec):

(Eq.2)http://en.wikipedia.org/wiki/Fil...Fig1. Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT).The integer k has units of cycles/sample, and1/T is the sample-rate, fs (samples/sec). SoX1/T(f) comprises exact copies of X(f) that are shifted by multiples of fs hertz and combined by addition. For sufficiently large fs the k=0 term can be observed in the region [−fs/2, fs/2] with little or no distortion (aliasing) from the other terms. In Fig.1, the extremities of the distribution in the upper left corner are masked by aliasing in the periodic summation (lower left).We also note that

is the Fourier transform of

Therefore, an alternative definition of DTFT is:[note1]

(Eq.3)The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.[1]Periodic data[edit]When the input data sequence x[n] is N-periodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT), because:

All the available information is contained within N samples.
converges to zero everywhere except integer multiples of known as harmonic frequencies.
The DTFT is periodic, so the maximum number of unique harmonic amplitudes is

The kernel

is N-periodic at the harmonic frequencies,

So

is an infinite summation of repetitious values, which does not converge for one or more values of k. But because of periodicity, we can reduce the limits of summation to any sequence of length N, without losing any information. The result is just a DFT. In order to interpret the DFT, it is helpful to expand the comb function, from Eq.3, which is now NT-periodic, into a Fourier series:

which also shows that periodicity in the time domain causes the DTFT to become discontinuous and that it diverges at the harmonic frequencies. But the Fourier series coefficients that modulate the comb are finite, and the standard integral formula conveniently reduces to a DFT:

which is an N-periodic sequence (in k) that completely describes the DTFT.Inverse transform[edit]An operation that recovers the discrete data sequence from the DTFT function is called aninverse DTFT. For instance, the inverse continuous Fourier transform of both sides ofEq.2 produces the sequence in the form of a modulated Dirac comb function:

However, noting that X1/T(f) is periodic, all the necessary information is contained within any interval of length1/T. In both Eq.1 and Eq.2, the summations over n are aFourier series, with coefficients x[n]. The standard formulas for the Fourier coefficients are also the inverse transforms:

Sampling the DTFT[edit]When the DTFT is continuous, a common practice is to compute an arbitrary number of samples (N) of one cycle of the periodic function X1/T:

where xN is a periodic summation:

The xN sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic.In order to evaluate one cycle of xN numerically, we require a finite-length x[n] sequence. For instance, a long sequence might be truncated by a window function of length Lresulting in two cases worthy of special mention: L ≤ N and L = I•N, for some integer I(typically6 or8). For notational simplicity, consider the x[n] values below to represent the modified values.When L = I•N a cycle of xN reduces to a summation of I blocks of length N. This goes by various names, such as multi-block windowing and window presum-DFT [2] [3][4] . A good way to understand/motivate the technique is to recall that decimation of sampled data in one domain (time or frequency) produces aliasing in the other, and vice versa. The xN summation is mathematically equivalent to aliasing, leading to decimation in frequency, leaving only DTFT samples least affected by spectral leakage. That is usually a priority when implementing an FFT filter-bank (channelizer). With a conventional window function of length L, scalloping loss would be unacceptable. So multi-block windows are created using FIR filter design tools. Their frequency profile is flat at the highest point and falls off quickly at the midpoint between the remaining DTFT samples. The larger the value of parameter I the better the potential performance. We note that the same results can be obtained by computing and decimating an L-length DFT, but that is not computationally efficient.When L ≤ N the DFT is usually written in this more familiar form:

In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all N terms, even though N-L of them are zeros. Therefore, the case L < N is often referred to as zero-padding.Spectral leakage, which increases as L decreases, is detrimental to certain important performance metrics, such as resolution of multiple frequency components and the amount of noise measured by each DTFT sample. But those things don't always matter, for instance when the x[n] sequence is a noiseless sinusoid (or a constant), shaped by a window function. Then it is a common practice to use zero-paddingto graphically display and compare the detailed leakage patterns of window functions. To illustrate that for a rectangular window, consider the sequence:

and

The two figures below are plots of the magnitude of two different sized DFTs, as indicated in their labels. In both cases, the dominant component is at the signal frequency: f =1/8 =0.. Also visible on the right is the spectral leakage pattern of the L= rectangular window. The illusion on the left is a result of sampling the DTFT at all of its zero-crossings. Rather than the DTFT of a finite-length sequence, it gives the impression of an infinitely long sinusoidal sequence. Contributing factors to the illusion are the use of a rectangular window, and the choice of a frequency (1/8 =8/) with exactly8 (an integer) cycles per samples.http://en.wikipedia.org/wiki/Fil...DFT for L = and N =http://en.wikipedia.org/wiki/Fil...DFT for L = and N =Convolution[edit]The Convolution theorem for sequences is:

An important special case is the circular convolution of sequences x and y defined byxN * y where xN is a periodic summation. The discrete-frequency nature of DTFT{xN} "selects" only discrete values from the continuous function DTFT{y}, which results in considerable simplification of the inverse transform. As shown atConvolution_theorem#Functions_of_a_discrete_variable..._sequences:

For x and y sequences whose non-zero duration is ≤ N, a final simplification is:

The significance of this result is expounded at Circular convolution and Fast convolution algorithms.Relationship to the Z-transform[edit]The bilateral Z-transform is defined by:

where z is a complex variable.On the unit circle, z is constrained to values of the form

Then one cycle of

is equivalent to one period of the DTFT. What varies with sample-rate is the width of a signal's spectral distribution. When the width exceeds2π, because of a sub-Nyquist rate, the distribution fills the circle, and aliasing occurs. With a DTFT in units of hertz (Eq.2), it's not the bandwidth that changes, but the periodicity of the aliases.Alternative notation[edit]The notation,

, is also often used to denote a normalized DTFT (Eq.1), which has several desirable features:

highlights the periodicity property, and
helps distinguish between the DTFT and the underlying Fourier transform of x(t); that is, X(f) (or X(ω)), and
emphasizes the relationship of the DTFT to the Z-transform.

However, its relevance is obscured when the DTFT is expressed as its equivalent periodic summation. So the notation X(ω) is also commonly used, as in the table below.Table of discrete-time Fourier transforms[edit]Some common transform pairs are shown in the table below. The following notation applies:

ω =2πfT is a real number representing continuous angular frequency (in radians per sample). (f is in cycles/sec, and T is in sec/sample.) In all cases in the table, the DTFT is2π-periodic (in ω).
X2π(ω) designates a function defined on -∞ < ω < ∞.
X(ω) designates a function defined on -π < ω ≤ π, and zero elsewhere. Then:

δ(ω) is the Dirac delta function
sinc(t) is the normalized sinc function
rect(t) is the rectangle function
tri(t) is the triangle function
n is an integer representing the discrete-time domain (in samples)
u[n] is the discrete-time unit step function
δ[n] is the Kronecker delta δn,0

Time domainx[n]Frequency domainX2π(ω)Remarks

integer M

odd M

even Minteger M >0

The

term must be interpreted as a distribution in the sense of aCauchy principal valuearound its poles at ω =2πk.

Written Dec

Saad Ahmed 2.9k Views Dft is for finite duration discrete periodic signals having a period say N= . It has a direct relation with dtft as a matter of fact dft is obtained by frequency sampling the spectrum of dtft. The continuous dtft spectrum when sampled at a particular frequency gives the dft spectrum which is discrete in the frequency domain. Written Nov • View Upvotes

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Determine the solution of the difference equation? Differentiate DTFT and DFT.

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