[DSP] W04 - Discrete Time Fourier Transform

content

• DFT
• DFS (Discrete Fourier Series)
  • finite-length time shifts
  • periodic-sequences
  • spectral information
• Discrete-Time Fourier Transform (DTFT)
  • dtft - formally
  • dtft periodicity and notation
• dtft examples
  • exponentially decaying sequence
  • infinite-support sawtooth
• existence and properties of DTFT
  • a formal change of basis
• review
  • DFT
  • DFS
  • DTFT
• DTFT properties
  • useful particular cases
  • DTFT as a change of basis
  • dirac-delta functional
• summary
• references

---

• to explore application of Fourier analysis to the three different classes of signals,
  • finite-length signals,
  • periodic signals and
  • infinite-length signals

---

DFT

• we have already looked in details at the case of finite-length signals where the DFT is the Fourier tool of choice

• an interesting property of the DFT:
  • if we let the DFT synthesis run beyond $N-1$, we obtain a $N$-periodic signal $x[n+N]=x\left[n\right]x[n+N]=x\left[n\right]$
• likewise, the analysis formula produces also a $N$-periodic series of Fourier coefficients
  • the concept of DFT naturally extends to periodic sequences
• discrete Fourier series (DFS) will be addressed in the case of periodic sequences
  • the Karplus-Strong algorithm will be revisited to illustrate a key point of this Fourier Representation

---

DFS (Discrete Fourier Series)

• DFS = DFT with periodicity explicit

• DFS maps:
  • an $N$-periodic signal onto
  • an $N$-periodic sequence of Fourier coefficients
• inverse DFS maps an:
  • $N$-periodic sequence of Fourier coefficients
  • a set onto an $N$-periodic signals
• the DFS of an $N$-periodic signal is mathematically equivalent to the DFT of one period

finite-length time shifts

• DFS helps us understand how to define time shifts for finite-length signals

• for an $N$-periodic sequence $\tilde{x}\left[n\right]$
  • $\tilde{x}[n-M]$ is well-defined for all $M\in N$
• DFS: $\left\{\tilde{x}\right[n-M\left]\right\}=e^{-j\frac{2\pi }{N}Mk}\tilde{X}\left[k\right]$
• Inverse DFS: $\left\{e^{-j\frac{2\pi }{N}Mk}\tilde{X}\right[k\left]\right\}=\tilde{x}[n-M]$
  • $e^{-j\frac{2\pi }{N}Mk}$: delay factor in frequency domain
• for an $N$-point signal $x\left[n\right]$:
  • $x[n-M]$ is not well-defined
  • build $\tilde{x}\left[n\right]=x\left[n\text{mod}N\right]\Rightarrow \tilde{X}\left[k\right]=X\left[k\right]$
  • mathematically, DFS of this signal is equal to the DFT of the original signal
• so, inverse DFT is the same as the inverse DFS of the same quantity
  • IDFT$\left\{e^{-j\frac{2\pi }{N}Mk}X\right[k\left]\right\}=$ IDFS $\left\{e^{-j\frac{2\pi }{N}Mk}X\right[k\left]\right\}$
  • $\tilde{x}[n-M]=x\left[\right(n-M\left)\right]\text{mod}N$
• shifts for finite-length signals are naturally circular
  • circular shifts are natural extension of shift operator
  • underlying fourier representation is the same as the one used for a periodic signal built on the finite-length signal

periodic-sequences

• $N$-periodic sequence: $N$ degrees of freedom

• DFS: only $N$ fourier coefficients capture all the information

• Karplus-Strong algorithm

  

  fig: Karplus-Strong algorithm

  • this can be used to generate a simple periodic signal
  • choose a finite-support signal $\overline{x}\left[n\right]$ that is non-zero only for $0\le n<M$
  • choose $\alpha =1$: no attenuation in loop
    • $y\left[n\right]=\overline{x}\left[0\right],\overline{x}\left[1\right],\dots ,\overline{x}[M-1],\overline{x}\left[0\right],\overline{x}\left[1\right],\dots ,\overline{x}[M-1],\overline{x}\left[0\right],\overline{x}\left[1\right],\dots$
    • $1^{st}$ period, $2^{nd}$ period

  

  fig: 32-tap sawtooth-wave input to Karplus-Strong algorithm

  • the output of the Karplus-Strong algorithm will be a period saw-tooth signal wave with the input saw-tooth as the input

  

  fig: DFT of output of Karplus-Strong algorithm

spectral information

• all the spectral information of a periodic signal is contained in the DFT of a single period
  • so a DFS is used for a periodic signal
• if we take the DFT of L repetitions of a finite length sequence of length $N$, we obtain a series which is non-zero only at multiple integers of L
• these non-zero coefficients are just scaled versions of the DFT coefficients of the original finite-length sequence
  • all the spectral information of an $N$-periodic sequence is entirely captured by the DFT coefficients of one period

---

discrete-time fourier transform (DTFT)

• the Fourier tool of choice for infinite length sequences
• derive an intuition from the Karplus-Strong algorithm
• we will then consider the DTFT more formally, in particular, studying conditions related to its existence and its properties
• it is a special type of basis expansion in the space of infinite sequences

fourier representation for signal classes

• $N$-point finite-length: DFT
• $N$-point periodic: DFS

• infinite length (non-periodic): DTFT

• Karplus-Strong to generate infinite-length:
  • set $\alpha <1$
  • generated signal is infinite-length but not periodic
  • $y\left[n\right]=\overline{x}\left[0\right],\overline{x}\left[1\right],\dots ,\overline{x}[M-1],$; $1^{st}$ period
    • $\alpha \overline{x}\left[0\right],\alpha \overline{x}\left[1\right],\dots ,\alpha \overline{x}[M-1],$ $2^{nd}$ period
    • ${\alpha }^2\overline{x}\left[0\right],{\alpha }^2\overline{x}\left[1\right],\dots$
• what is a good spectral representation for infinite-length signals?
  • start with DFT and asses what happens when the signal length grows towards $\infty$
    • $\frac{2\pi }{N}k\text{becomes denser in}[0,2\pi ]$
  • in the limit $\frac{2\pi }{N}k\to \omega$:
    • ${\sum }_{n}x\left[n\right]e^{-j\omega n}$, $\omega \in R$

dtft - formally

• $x\left[n\right]\in {\ell }_2\left(Z\right)$: square-summable i.e. finite energy
• define the function of $\omega \in R$ (not the series itself)
  • $F\left(\omega \right)={\sum }_{n=-\infty }^{\infty }x\left[n\right]e^{-j\omega n}$
• inversion (when $F\left(\omega \right)$ exists:
  • $x\left[n\right]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F\left(\omega \right)e^{j\omega n}d\omega$; $n\in Z$

dtft periodicity and notation

• $F\left(\omega \right)$ is $2\pi$-periodic
• to stress periodicity, it is written so:
  • $X\left(e^{j\omega }\right)={\sum }_{n=-\infty }^{\infty }x\left[n\right]e^{-j\omega n}$
• by convention, $X\left(e^{j\omega }\right)$ is represented over $[-\pi ,\pi ]$ for DTFT

---

dtft examples

exponentially decaying sequence

• consider a exponentially decaying example

  

  fig: exponentially decaying sequence - $x\left[n\right]={\alpha }^{n}u\left[n\right],\left|\alpha \right|<1$

  • this sequence is infinite and non-periodic

• applying DTFT to this: $$X\left(e^{j\omega }\right)=\sum _{n=-\infty }^{\infty }x\left[n\right]e^{-j\omega n}$$ $$=\sum _{n=0}^{\infty }{\alpha }^n{e}^{-j\omega n}$$ $$=\sum _{n=0}^{\infty }(\alpha e^{-j\omega }{)}^n$$ $$=\frac{1}{1-\alpha e^{(-j\omega )}}$$

• magnitude of the fourier transform: $$\left|X\right(e^{j\omega }){|}^2=\frac{1}{1+{\alpha }^2-2\alpha \cos \omega }$$

• plotting DTFT:

  

  fig: dtft of $x\left[n\right]={\alpha }^{n}u\left[n\right],\left|\alpha \right|<1$

• most energy is consolidated around the low-frequencies
  • it moves slowly to zero
  • remember the inherent fourier transform
  • if the frequency axis is expanded out of the $[-\pi ,\pi ]$ range, the signal repeats with a periodicity of $2\pi$

  

  fig: periodicity of dtft

• DTFT frequency distributions

  

  fig: positive frequencies

  

  fig: negative frequencies

  

  fig: slow and fast frequencies

infinite-support sawtooth

• generate first an infinite-support non-periodic signal
  • then compute spectrum with DTFT paradigm
• the Karplus-Strong algorithm will be used to generate the non-periodic infinite length signal
  • a sawtooth input is used as the initial data

fig: input signal for Karplus-Strong algorithm

• the output of the Karplus-Strong algorithm is shown below

fig: output of the Karplus-Strong algorithm (exponentially-decaying saw-tooth wave), $\alpha <1$

• DTFT of this exponentially-decaying saw-tooth wave

$Y\left(e^{j\omega }\right)={\sum }_{n=-\infty }^{\infty }y\left[n\right]e^{-j\omega n}={\sum }_{p=0}^{\infty }{\sum }_{n=0}^{M-1}{\alpha }^p\overline{x}\left[n\right]e^{-j\omega (pM+n)}={\sum }_{p=0}^{\infty }{\alpha }^p{e}^{-j\omega Mp}{\sum }_{n=0}^{M-1}\overline{x}\left[n\right]e^{-j\omega (pM+n)}=A\left(e^{j\omega M}\right)\overline{X}\left(e^{j\omega }\right)$

• this indicates a rescaling of the frequency axis

• DTFT of the exponential decay output of the Karplus-Strong algorithm

  • the first and the second term of the product are dealt with separately

  • first term: $A\left(e^{j\omega M}\right)$:

  

  fig: DTFT of KS-algorithm output; M = 1; first term of product

  

  fig: DTFT of KS-algorithm output; M = 2; first term of product

  

  fig: DTFT of KS-algorithm output; M = 3; first term of product

  

  fig: DTFT of KS-algorithm output; M = 12; first term of product

  • second term: $\overline{X}\left(e^{j\omega }\right)$
    • $=e^{j\omega }(\frac{M+1}{M-1})\frac{1-e^{-j(M-1)\omega }}{(1-e^{-j\omega }{)}^2}-\frac{1-e^{-j(M+1)\omega }}{(1-e^{-j\omega }{)}^2}$

  

  fig: DTFT of KS-algorithm output - second term of product

  • final spectrum: $Y\left(e^{j\omega }\right)=A\left(e^{j\omega M}\right)\overline{X}\left(e^{j\omega }\right)$

  

  fig: DTFT of KS-algorithm output - second term of product

  • this is the DTFT (spectrum) of an infinite non-periodic signal that is not trivial
  • peaks of the first term in the product slim down the lobes of the second term

• here DTFT is treated as a signal operator

  • it is a function of a real variable $\omega$
  • DTFT is NOT an extension of the DFT for finite-length signals
    • it is it’s own paradigm for infinite-length signals
  • DTFT is $2\pi$ periodic, hence it is written as a function of $e^{j\omega }$

---

existence and properties of DTFT

• existence simply means the sum that defines the dtft does not blow up

  • easy to prove for absolutely summable sequences

    $\left|X\right(e^{j\omega }\left)\right|=\left|{\sum }_{n=-\infty }^{\infty }x\right[n\left]e^{-j\omega n}\right|\le {\sum }_{n=-\infty }^{\infty }\left|x\right[n\left]e^{-j\omega n}\right|={\sum }_{n=-\infty }^{\infty }\left|x\right[n\left]\right|<\infty$

• if we assume absolute summability of the underlying sequence
  • we can invert the DTFT

  • begin with the inversion formula on LHS

    $\frac{1}{2\pi }{\int }_{\pi }^{pi}X\left(e^{j\omega }\right)e^{j\omega n}d\omega =\frac{1}{2\pi }{\int }_{\pi }^{\pi }({\sum }_{k=-\infty }^{\infty }x\left[k\right]e^{-j\omega k})e^{j\omega n}d\omega ={\sum }_{k=-\infty }^{\infty }x\left[k\right]{\int }_{\pi }^{\pi }\frac{e^{j\omega (n-k)}}{2\pi }d\omega =x\left[n\right]$

a formal change of basis

• a DTFT looks exactly like an inner-product in $C^{\infty }$ space $$\sum _{n=-\infty }^{\infty }x\left[n\right]e^{-j\omega n}=⟨e^{j\omega n},x\left[n\right]⟩$$

• $C^{\infty }$ is not a well-defined vector space
  • there are many sequences that do not converge in this space
• but, if a formal parallel between a change of basis and the DTFT is established
  • then everything discovered about the DFT (which is well defined) will apply to the DTFT as well
• here, the DTFT basis is not really a basis, because it is an infinite and uncountable set of vectors
  • indexed by a real values variable $\omega$
  • $\{e^{j\omega n}{\}}_{\omega \in R}$
• something breaks down
  • start with sequences but the transformation is a function
  • DTFT exists for all square-summable sequences (larger set of sequences)
    • but absolutely summable sequences was used

---

review

• comparison of DFT, DFS and DTFT
  • change of basis
  • basis vector set
  • inversion
  • time and frequency domains

DFT

• for finite-length signals

  

  fig: DFT: change of basis from original sequence in time domain to frequency domain and back via inversion

• here, the time domain sequence and the corresponding frequency domain entity after change of basis lives in $C^N$
• the basis which allows going from time domain to frequency domain is the DFT basis
  • this is a countable set for $N$ fourier basis vectors

DFS

• for periodic signals

  

  fig: DFS: change of basis from a periodic in time domain to frequency domain and back via inversion

• the expansion and reconstruction formulas are the same
• except in this case we assume everything is periodic

DTFT

• for infinite-length signals

  

  *fig: DTFT: change *

• start from space of square-summable sequences ${\ell }_2\left(Z\right)$, through a “formal” change of basis,

  • the original sequence is projected onto a space of square integrable functions $L_2[-\pi ,\pi ]$
  • on the interval $[-\pi ,\pi ]$

---

DTFT properties

• linearity follows from the properties of the inner-product $$DTFT\left\{\alpha x\right[n]+\beta y[n\left]\right\}=\alpha X\left(e^{j\omega }\right)+\beta Y\left(e^{j\omega }\right)$$
  • DTFT of a linear combination of two sequences will be linear combination of the DTFT of the sequences
    
• time-shift: $$DTFT\left\{x\right[n-M\left]\right\}=e^{-j\omega M}X\left(e^{j\omega }\right)$$
  • the DTFT of a sequence shifted by $M$ is the DTFT of the original sequence scaled by a delay factor
    • delay factor: $e^{-j\omega M}$
      
• modulation: $$DTFT\left\{e^{j{\omega }_{0}n}x\right[n\left]\right\}=X\left(e^{j(\omega -{\omega }_o)}\right)$$
  • if we take a sequence and multiply it with a complex exponential at frequency ${\omega }_0$ and take the DTFT of that,
    • that is same as applying a shift of ${\omega }_0$ to the spectrum obtained by DTFT
      
• time-reversal $$DTFT\left\{x\right[-n\left]\right\}=X\left(e^{-j\omega }\right)$$
  • fourier transform of a time reversed sequence (values flipped about the origin) is equal to a frequency reversed Fourier Transform
    
• conjugation $$DTFT\left\{x^{\ast }\right[n\left]\right\}=X^{\ast }\left(e^{-j\omega }\right)$$
  • if every value of the sequence is conjugated, the fourier transform will be both conjugated and frequency reversed
    

useful particular cases

• if a signal sequence $x\left[n\right]$ is symmetric, the DTFT is symmetric $$x\left[n\right]=x[-n]\iff X\left(e^{j\omega }\right)=X\left(e^{-j\omega }\right)$$

• if sequence $x\left[n\right]$ is real, DTFT is Hermitian-symmetric $$x\left[n\right]=x^{\ast }\left[n\right]\iff X\left(e^{j\omega }\right)=X^{\ast }\left(e^{-j\omega }\right)$$
  • reality of the sequence can be expressed mathematically by $x\left[n\right]=x^{\ast }\left[n\right]$
  • this in frequency domain implies that fourier transform of the sequence is equal to the conjugate and frequency reversed version of the fourier transform
• special case (corollary) of above:
  • if $x\left[n\right]$ is real, the magnitude of the DTFT is symmetric $$x\left[n\right]\in R\Rightarrow \left|X\right(e^{j\omega }\left)\right|=X\left(e^{-j\omega }\right)$$
• a more special case of above:
  • if $x\left[n\right]$ is real and symmetric,
  • $X\left(e^{j\omega }\right)$ is also real and symmetric
• this mostly looks like a full-fledged basis expansion
  • so DTFT is just another version of the DFT
  • considering all the particular cases

DTFT as a change of basis

• comparing DFT and DTFT, some operations are acceptable
  • $DFT\left\{\delta \right[n\left]\right\}=1$
  • $DTFT\left\{\delta \right[n\left]\right\}=⟨e^{j\omega n},\delta \left[n\right]⟩=1$
• other operations don’t work;
  • $DFT\left\{1\right\}=N\delta \left[k\right]$
  • $DTFT\left\{1\right\}={\sum }_{n=-\infty }^{\infty }{e}^{-j\omega n}=?$
• a lot of interesting sequences are not square summable
  • this is addressed by introducing the Dirac-delta functional

dirac-delta functional

• “sifting” property: $${\int }_{-\infty }^{\infty }\delta (t-s)f\left(t\right)dt=f\left(s\right)$$ for all functions of $t\in R$

  • $s\in R$ along axis of $t$

  • dirac-delta is a function which is an upward pointing arrow at $t=s$
  • consider any function $f\left(t\right)$, multiply this dirac-delta with $f\left(t\right)$
  • integrate the multiplied function from $-\infty$ to $\infty$
    • this yields the value of $f\left(t\right)$ @ $t=s$

• to develop intuition for the dirac-delta function
  • consider a family of localizing functions $r_k\left(t\right)$ with $k\in N$ and $t\in R$
• the properties of these functions are:
  • support of each function is inversely proportional to it’s index $k$
  • integral of each function from $-\infty$ to $\infty$ is constant
    • i.e. the area under each function is constant, regardless of index $k$
• $rect\left(t\right)=\{\begin{array}{cc}1& \text{for}\left|t\right|<\frac{1}{2}\\ 0& \text{otherwise}\end{array}$
• consider the localizing family $r_k\left(t\right)=k\text{rect}\left(kt\right)$:
  • nonzero over $[-\frac{1}{2k},\frac{1}{2k}]$ i.e. support is $\frac{1}{k}$
  • area is $1$

  

  fig: k = 1, rect function

  

  fig: k = 5, rect function

  

  fig: k = 15, rect function

  

  fig: k = 40, rect function

• extracting a point value:
  • consider the integral of the product of one of these rect functions and any function of real time

  $${\int }_{-\infty }^{\infty }{r}_k\left(t\right)f\left(t\right)dt=k{\int }_{-\frac{1}{2k}}^{\frac{1}{2k}}f\left(t\right)dt=f\left(\gamma \right){|}_{\gamma \in [-\frac{1}{2k},\frac{1}{2k}]}$$

  • $r_k\left(t\right)$ is non-zero only between $[-\frac{1}{2k},\frac{1}{2k}]$
  • so, integration bounds change from $[-\infty ,\infty ]$ to $[-\frac{1}{2k},\frac{1}{2k}]$
  • then apply mean-value theorem to get $=f\left(\gamma \right){|}_{\gamma \in [-\frac{1}{2k},\frac{1}{2k}]}$
    • value of $\gamma$ is not known, but it is known to exist
  • as $k\to \infty$, the support of the $r_k$ gets smaller and smaller $$\underset{k\to \infty }{lim}{\int }_{-\infty }^{\infty }{r}_k\left(t\right)f\left(t\right)dt=f\left(0\right)$$
  • see above plots to get an intuitive feel
  • as $k$ increases, the width of the support shrinks to an infinitesimal width
    • so, at the limit, the limit value is $f\left(0\right)$
  • the dirac-delta function is the shorthand for this limiting operation
    • it uses the concept of narrowing support functions as height of the support function increases to obtain the value of a function at a point in time
  • the delta functional is a shorthand: $$\underset{k\to \infty }{lim}{\int }_{-\infty }^{\infty }{r}_k(t-s)f\left(t\right)dt$$
    • is instead written as $${\int }_{-\infty }^{\infty }\delta (t-s)f\left(t\right)dt$$
    • as if ${lim}_{k\to \infty }{r}_k\left(t\right)=\delta \left(t\right)$
• the “pulse train”
  • the dirac-delta functional will be used in the frequency domain
  • all DTFT spectra are $2\pi$ periodic
  • to be able to use the dirac-delta functional in the frequency domain, it has be periodized
    • this periodized version of the dirac-delta functional is called the “pulse train”
  • this done by placing copies of the dirac-delta functional every $2\pi$ and scaling the whole thing by $2\pi$ $$\tilde{\delta }\left(\omega \right)=2\pi \sum _{k=-\infty }^{\infty }\delta (\omega -2\pi k)$$

  

  fig: dirac delta pulse-train in frequency domain

• consider the inverse DTFT of the pulse train: $$IDTFT\left\{\tilde{\delta }\right(\omega \left)\right\}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\tilde{\delta }\left(\omega \right)e^{j\omega n}d\omega ={\int }_{-\pi }^{\pi }\tilde{\delta }\left(\omega \right)e^{j\omega n}d\omega =e^{j\omega n}{|}_{\omega =0}=1$$

• comparing with IDFT:
  • $IDFT\left\{N\delta \right[k\left]\right\}=1$

• so, $DTFT\left\{1\right\}=\tilde{\delta }\left(\omega \right)$

• more identities using dirac-delta pulse train results

  • $IDTFT\left\{\tilde{\delta }\right(\omega -{\omega }_0\left)\right\}=e^{j{\omega }_{0}n}$

  • $DTFT\left\{1\right\}=\tilde{\delta }\left(\omega \right)$
  • $DTFT\left\{e^{j{\omega }_{0}n}\right\}=\tilde{\delta }(\omega -{\omega }_0)$
  • $DTFT\left\{\cos \right({\omega }_{0}n\left)\right\}=\frac{\left[\tilde{\delta }\right(\omega -{\omega }_0)+\tilde{\delta }(\omega +{\omega }_0\left)\right]}{2}$
  • $DTFT\left\{\sin \right({\omega }_{0}n\left)\right\}=-j\frac{\left[\tilde{\delta }\right(\omega -{\omega }_0)-\tilde{\delta }(\omega +{\omega }_0\left)\right]}{2}$

---

summary

• the fourier tool of choice for infinite-length sequences, called the discrete-time fourier transform (DTFT) has been explored
• Karplus-Strong algorithm was used to derive intuition; an infinite sequence was generated by repeating a finite length sequence and scaling each repetition by an exponential decay factor
  • as the number of repetitions increased, the fundamental frequencies $\frac{2\pi }{N}k$ become denser and denser
  • this can be replaced in the DFT formula with a real frequency $\omega \in R$
  • this leads to the definition of the DTFT analysis and synthesis formulas

$$X\left(e^{j\omega }\right)=\sum _{n=-\infty }^{+\infty }x\left[n\right]e^{-j\omega n}$$

$$x\left[n\right]=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }X\left(e^{j\omega }\right)e^{j\omega n}d\omega$$

• the DTFT thus maps an infinite length sequence $x\left[n\right]$ onto a function of real variable $\omega$
• the DTFT is $2\pi$-periodic
  • is expressed as$X\left(e^{j\omega }\right)$ to stress this, although only $\omega$ is the variable
  • this is standard convention in the DTFT literature
  • so, furthermore, by convention, DTFT is represented on the $[-\pi ,\pi ]$ interval
• frequencies between 0 and $\pi$ (resp. $-\pi$ and 0) are the positive (resp. negative) frequencies, corresponding to counterclockwise (resp. clockwise) rotations

• low (resp. high) frequencies are those close to 0 (resp. $-\pi$ and $\pi$ )

• the question of existence is essential to ensure that the sum of the analysis formula does not blow up

• tt can be shown that the DTFT converges for all square summable sequences

• DTFT can be interpreted as a change of basis in the space $C^{\infty }$ where the uncountable set of functions $e^{j\omega n}\omega \in R$ forms an orthogonal basis
• this space is not a proper vector space and a mathematical adjustment is needed to reconcile this intuition
• hence the concept of the dirac-delta function is introduced
  • $\delta \left(t\right)$ of real variable $t$ is defined by the “sifting property” $${\int }_{-\infty }^{\infty }\delta (t-s)f\left(t\right)dt=f\left(s\right);\forall f\left(t\right),t\in R$$
• with this adjustment, the DTFT can be defined for certain interesting non square summable sequences, such as

$$\text{DTFT}\left(1\right)=2\pi \sum _{k=-\infty }^{\infty }\delta (w-2\pi k)$$

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references

• ecole dsp - wk 4