# DFT and FFT

### 4. DFT and convolution

Let $$h(n)$$ be the sequence $$\{1,1,0,0,0,0,0,0\}$$ and $$y(n)=\{1,1,1,1,0,0,0,0\}$$.

1. Calculate the DFT of length $$8$$ for both sequences.
2. Determine with help of the DFT a sequence $$v(n)$$ such that $$y(n)= h(n) \circledast v(n)$$.
3. Let $$z(n)$$ be the result of the linear convolution of $$h(n)$$ and $$v(n)$$: $$z(n)=h(n)*v(n)$$. Is $$z(n)=y(n)$$?

## Amount and difficulty

• Working time: approx. 15 minutes
• Difficulty: easy

### 5. DFT

The time-limited signal

$$v_0(t)=\left\{\begin{array}{ll} \sin(\omega_0 t) & for\quad 0\le t< 4\pi/\omega_0 \\ 0 & otherwise \end{array}\right.$$

is sampled with $$t_n=n T_{\textrm{A}}=n\frac{ \pi}{4 \omega_0}$$ to produce the time-limited sequence $$v(n)$$ .

1. Sketch $$v_0(t)$$.
2. Determine the DFT of $$v(n)$$.
3. Determine the Fourier Transform $$V(e^{j\Omega})$$ of $$v(n)$$.
4. Explain the connection between the DFT$$\{v(n)\}$$ and $$V(e^{j\Omega})$$.

## Amount and difficulty

• Working time: approx. 20 minutes
• Difficulty: easy

### 6. DFT, zero padding, leakage

Let $$v_{\textrm{a}}(t)$$ be a time-continuous periodic signal $$v_{\textrm{a}}(t)= 1+ cos(2\pi 40t) + 3\cdot cos(2\pi 120t). \nonumber$$ The signal is sampled ($$\omega_s = 2\pi 280 s^{-1}$$) to produce the sequence $$v(n)$$. For practical purposes (delay, complexity) the sequence is limited to $$L$$ samples. $$M$$ is the length of the DFT. Use MATLAB to solve the following subproblems.

1. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=7$$ and $$M=7$$.
2. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=7$$ and $$M=14$$ (zero padding).
3. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=28$$ and $$M=28$$.
4. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=7$$ and $$M=7$$.
5. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=28$$ and $$M=56$$ (zero padding).
6. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=14$$ and $$M=15$$ (zero padding).
7. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=14$$ and $$M=21$$ (zero padding).
8. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=30$$ and $$M=30$$.
9. Sketch $$v_{\textrm{a}}(t)$$, $$v(n)$$, the Fourier transform $$V(e^{j\Omega})$$ and the DFT $$V_M(\mu)$$ for $$L=15$$ and $$M=30$$ (zero padding).

## Amount and difficulty

• Working time: approx. 25 minutes
• Difficulty: middle

### 7. FFT

Let $$v(n)$$ be a time-discrete signal $$v(n) = [v(0), v(1), v(2), v(3), v(4), v(5), v(6), v(7)]$$.

1. Separate the signal $$v(n)$$ into even and odd time-indices $$v_1(n)$$ and $$v_2(n)$$ respectively and find the DFT expression for each separated sequence.
2. Now compute the DFT of $$v(n)$$ using the above expressions.
3. Sketch the signal flow diagrams when DFT is directly applied to $$v(n)$$ and as shown in part (b). Show the reduction in complexity by computing the number of complex multiplications for each method.
4. Can the complexity be reduced further? If yes then find the final expression.
5. Sketch the complete signal flow for part (d).

## Amount and difficulty

• Working time: approx. 15 minutes
• Difficulty: middle

### 8. FFT

The $$M$$-point DFT of the $$M$$-point sequence $$x(n) = e^{-j(\pi/M)n^2}$$, for $$M$$ even, is

$$X(\mu) = \sqrt{M}e^{-j\pi/4}e^{j(\pi/M)\mu^2}$$.

Determine the $$2M$$-point of sequence $$y(n) = e^{-j(\pi/M)n^2}$$, assuming that $$M$$ is even.

## Amount and difficulty

• Working time: approx. 15 minutes
• Difficulty: middle

### 9. FFT of real and complex sequences

Suppose that an FFT program is available that computes the DFT of a complex sequence. If we wish to compute the DFT of a real sequence, we may simply specify the imaginary part to be zero and use the program directly. However, the symmetry of the DFT of a real sequence can be used to reduce the amount of computation.

1. Let $$x(n)$$ be a real-valued sequence of length $$M$$, and let $$X(\mu)$$ be its DFT with real and imaginary parts denoted $$X_R(\mu)$$ and $$X_I(\mu)$$, respectively; i.e.,

$$X(\mu) = X_R(\mu) + j\,X_I(\mu)$$.

Show that if $$x(n)$$ is real, then $$X_R(\mu) = X_R(M - \mu)$$ and $$X_I(\mu) = -X_I(M - \mu)$$ for $$\mu = 1,...,M-1$$.
2. Now consider two real-valued sequences $$x_1(n)$$ and $$x_2(n)$$ with DFTs $$X_1(\mu)$$ and $$X_2(\mu)$$, respectively. Let $$g(n)$$ be the complex sequence $$g(n) = x_1(n) + j\,x_2(n)$$, with corresponding DFT $$G(\mu) = G_R(\mu) + j\,G_I(\mu)$$. Also, let $$G_{OR}(\mu)$$, $$G_{ER}(\mu)$$, $$G_{OI}(\mu)$$ and $$G_{EI}(\mu)$$ denote, respectively, the odd part of the real part, the even part of the real part, the odd part of the imaginary part, and the even part of the imaginary part of $$G(\mu)$$. Specifically, for $$1 \leq \mu \leq M-1$$,

$$G_{OR}(\mu) = 1/2\{G_R(\mu) - G_R(M - \mu)\}$$,

$$G_{ER}(\mu) = 1/2\{G_R(\mu) + G_R(M - \mu)\}$$,

$$G_{OI}(\mu) = 1/2\{G_I(\mu) - G_I(M - \mu)\}$$,

$$G_{EI}(\mu) = 1/2\{G_I(\mu) + G_I(M - \mu)\}$$,

and $$G_{OR}(0) = G_{OI}(0) = 0$$, $$G_{ER}(0) = G_{R}(0)$$, $$G_{EI}(0) = G_{I}(0)$$. Determine expressions for $$X_1(\mu)$$ and $$X_2(\mu)$$ in terms of $$G_{OR}(\mu)$$, $$G_{ER}(\mu)$$, $$G_{OI}(\mu)$$ and $$G_{EI}(\mu)$$.

## Amount and difficulty

• Working time: approx. 15 minutes
• Difficulty: middle

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### Recent Publications

J. Reermann, P. Durdaut, S. Salzer, T. Demming,A. Piorra, E. Quandt, N. Frey, M. Höft, and G. Schmidt: Evaluation of Magnetoelectric Sensor Systems for Cardiological Applications, Measurement (Elsevier), ISSN 0263-2241, https://doi.org/10.1016/j.measurement.2017.09.047, 2017

S. Graf, T. Herbig, M. Buck, G. Schmidt: Low-Complexity Pitch Estimation Based on Phase Differences Between Low-Resolution Spectra, Proc. Interspeech, pp. 2316 -2320, 2017

### Contact

Prof. Dr.-Ing. Gerhard Schmidt

E-Mail: gus@tf.uni-kiel.de

Christian-Albrechts-Universität zu Kiel
Faculty of Engineering
Institute for Electrical Engineering and Information Engineering
Digital Signal Processing and System Theory

Kaiserstr. 2
24143 Kiel, Germany

## Recent News

Jens Reermann Defended his Dissertation with Distinction

On Friday, 21st of June, Jens Reermann defended his research on signals processing for magnetoelectric sensor systems very successfully. After 90 minutes of talk and question time he finished his PhD with distinction. Congratulations, Jens, from the entire DSS team.

Jens worked for about three and a half years - as part of the collaborative research center (SFB) 1261 - on all kinds of signal ...