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### Digital Signal Processing

Signal
digitized signal
Time
• By time sampling and quantization (rounding), an analog signal can be converted into a digital signal
• The advantage of digital signals is that they can be easily processed with a digital signal processor or a computer

### Discrete-Time Signals by Sampling

$x(t)$
$x[n]$
Zeit $t$
• Mathematically expressed: The analog time signal $\mathrm{x}(t)$ is converted during sampling into a discrete-time signal $\mathrm{x}[n]$ with $n \in \mathbb{Z}$
• Since $n$ is an element from the set of integers, the function $\mathrm{x}[n]$ is always discrete in time
• A more detailed description of the digitization process follows in later chapters

## Discrete-Time Signals

### Discrete-Time Signals

• Sampling of analog signals is only one way to generate discrete-time signals
• Of course, any discrete-time signal can be generated directly in the computer
• Here are some of the most important discrete-time signals from a digital signal processing perspective

### Discrete-Time Signals: Unit Impulse

• Of very great importance is the so-called discrete-time unit impulse sequence (or "unit impulse" for short)
$\mathrm{\delta}[n] = \begin{cases} 1 & \,\,:\,\, n = 0 \\ 0 & \,\,:\,\, n \ne 0 \\ \end{cases}$
$\mathrm{\delta}[n]$
$n$
• The notation with $\mathrm{\delta}[n]$ is chosen because the equivalent signal in analog signal processing is the Dirac delta function $\mathrm{\delta}(t)$

### Discrete-Time Signals: Unit Step

• The discrete-time unit step $\mathrm{u}[n]$ is defined by:
$\mathrm{u}[n] = \begin{cases} 1 & \,\,:\,\, n \ge 0 \\ 0 & \,\,:\,\, n \lt 0 \\ \end{cases}$
$\mathrm{u}[n]$
$n$
• Is there a mathematical relationship between the unit step and the unit impulse?
• Yes, the integration (or in the discrete-time case, the summation):
$\mathrm{u}[n] = \sum\limits_{i=-\infty}^n \mathrm{\delta}[i]$

### Discrete-Time Signals: Rectangular Pulse

• A discrete-time rectangular pulse with pulse width $P$ is generated by:
$\mathrm{x}[n] = \begin{cases} 1 & \,\,:\,\, |n| < P/2 \\ 0.5 & \,\,:\,\, |n|=P/2 \\ 0 & \,\,:\,\, |n|> P/2 \\ \end{cases}$
• The figure shows a rectangular pulse with a pulse width $P=9$:
$\mathrm{x}[n]$
$n$
• The case $|n| = P/2$ can only occur for even $P$, e.g. $P=10$. In this case the value $0.5$ ensures that the pulse width is still $P$.

### Discrete-Time Signals: Gaussian Pulse

• A discrete-time Gaussian pulse with the standard deviation $\sigma$ is generated by:
$\mathrm{x}[n] = e^{- 0.5 \, (n / \sigma)^2}$
• The figure shows a Gauss pulse with standard deviation $\sigma=4$:
$\mathrm{x}[n]$
$n$

### Discrete-Time Signals: Triangle Pulse

• A time-discrete triangle pulse with pulse width $P$ is generated by:
$\mathrm{x}[n] = \begin{cases} 1.0 - 2.0 \, (n / P) & \,\,:\,\, |n| \le P/2 \\ 0 & \,\,:\,\, |n| > P/2 \\ \end{cases}$
• The figure shows a triangle pulse with a pulse width $P=9$:
$\mathrm{x}[n]$
$n$

### Discrete-Time Signals: Sine Wave

• A discrete-time sine signal can be generated as follows:
$\mathrm{x}[n] = A \sin\left(2\pi\frac{n+M}{W}\right)$
• The figure shows a sine wave with wavelength $W=16$, displacement $M=0$, and amplitude $A=1$:
$\mathrm{x}[n]$
$n$

### Discrete-Time Signals: Triangle Wave

• A discrete-time triangle wave can be generated by:
$\mathrm{x}[n] = A \left(2.0 \frac{(n+M) \, \bmod \, W}{W} - 1.0\right)$
where $\bmod$ denotes the modulo operation.
• The figure shows a triangle wave with wavelength $W=16$, displacement $M=0$, and amplitude $A=1$:
$\mathrm{x}[n]$
$n$

### Discrete-Time Signals: Square Wave

• A time-discrete square wave can be generated by:
$\mathrm{x}[n] = \begin{cases} A & :\quad (n+M) \, \bmod \, W \ge W /2 \\ -A & :\quad (n+M) \, \bmod \, W < W /2\\ \end{cases}$
• The figure shows a square wave for wavelength $W=16$, displacement $M=0$, and amplitude $A=1$:
$\mathrm{x}[n]$
$n$

## Convolution

### Convolution

• The discrete-time convolution of two signals $\mathrm{a}[n]$ and $\mathrm{b}[n]$ is defined by:
$\mathrm{f}[n] = \sum\limits_{k = -\infty}^{\infty} \mathrm{a}[k] \, \,\mathrm{b}[n-k]$
• To calculate $\mathrm{f}[n]$, the sum must be evaluated for different $n$. To this end, the signal $\mathrm{b}$ is mirrored at the $y$-axis and then shifted by $n$ to the right.
• This means, the sum is calculated for different offsets $n$ of the mirrored signal $\mathrm{b}$.
• For the short notation of the convolution, the operator "$\ast$" is used:
$\mathrm{f}[n] = \mathrm{a}[n] \ast \mathrm{b}[n] = \sum\limits_{k = -\infty}^{\infty} \mathrm{a}[k] \, \,\mathrm{b}[n-k]$

### Properties of the Convolution

• Commutativity
$\mathrm{a}[n] \ast \mathrm{b}[n] = \mathrm{b}[n] \ast \mathrm{a}[n]$
• Associativity
$\mathrm{a}[n] \ast (\mathrm{b}[n] \ast \mathrm{c}[n]) = (\mathrm{a}[n] \ast \mathrm{b}[n] ) \ast \mathrm{c}[n]$
• Distributivity
$\mathrm{a}[n] \ast (\mathrm{b}[n] + \mathrm{c}[n]) = \mathrm{a}[n] \ast \mathrm{b}[n] + \mathrm{a}[n] \ast \mathrm{c}[n]$

### Example: Convolution

$\mathrm{f}[n] = \sum\limits_{k = -\infty}^{\infty} \mathrm{a}[k] \, \,\mathrm{b}[n-k]$

$\mathrm{a}[k]$
$\mathrm{b}[k]$
$\mathrm{c}[n]$

### Example: Convolution

$\mathrm{f}[n] = \sum\limits_{k = -\infty}^{\infty} \mathrm{a}[k] \, \,\mathrm{b}[n-k]$

Konstruktion:

$\mathrm{a}[k]$
$\mathrm{b}[0-k]$
$\mathrm{a}[k]$
$\mathrm{b}[1-k]$
$\mathrm{f}[0]$
$\mathrm{f}[n]$
$\mathrm{f}[1]$
$\mathrm{f}[n]$

### Example: Convolution

$\mathrm{f}[n] = \sum\limits_{k = -\infty}^{\infty} \mathrm{a}[k] \, \,\mathrm{b}[n-k]$

Konstruktion:

$\mathrm{b}[5-k]$
$\mathrm{b}[7-k]$
$\mathrm{f}[5]$
$\mathrm{f}[7]$

## Impulse Response

### Linear Time-Invariant Systems

• Let us consider a system that transforms a discrete-time input sequence $\mathrm{x}[n]$ into the output sequence $\mathrm{y}[n]$ by a transfer function $\mathrm{S}(\mathrm{x}[n])$
$\mathrm{x}[n]$
$\mathrm{y}[n]$
System
$\mathrm{S}$
• If the system is a linear time-invariant system (LTI system), the impulse response together with the convolution operation is sufficient to describe the system completely

### Linear Time-Invariant Systems

• Linear time invariant systems (LTI systems) have the following defining properties:
• Linearity: Let $\mathrm{y}[n]_k$ be the reaction of the system to the input sequence $\mathrm{x}[n]_k$. Then it must applies to the linear combination of the input signals
$\mathrm{x}[n] = s_1 \, \mathrm{x}[n]_1 + s_2 \, \mathrm{x}[n]_2 + s_3 \, \mathrm{x}[n]_3 + \dots$
that the reaction of the system is
$\mathrm{y}[n] = s_1 \, \mathrm{y}[n]_1 + s_2 \, \mathrm{y}[n]_2 + s_3 \, \mathrm{y}[n]_3 + \dots$
or written differently with $\mathrm{y}[n]_k = \mathrm{S}\left(\mathrm{x}[n]_k\right)$:
$\mathrm{S}\left(\sum\limits_k \, s_k \mathrm{x}[n]_k\right) = \sum\limits_k s_k \, \mathrm{S}\left(\mathrm{x}[n]_k\right)$
• Time invariance: A shift of the input sequence by $M$ always results in a corresponding shift in the output sequence:
$\mathrm{S}\left(\mathrm{x}[n+M]\right) = \mathrm{y}[n+M] \quad \forall \,M \in \mathbb{Z}$

### Impulse Response

• The impulse response $\mathrm{h}[n]$ of an LTI system is defined as the response to the unit impulse $\mathrm{\delta}[n]$:
$\mathrm{h}[n] = \mathrm{S}\left(\mathrm{\delta}[n])\right)$
with
$\mathrm{\delta}[n] = \begin{cases} 1 & \,\,:\,\, n = 0 \\ 0 & \,\,:\,\, n \ne 0 \\ \end{cases}$
• For example:
$n$
$n$
$\mathrm{\delta}[n]$
$\mathrm{h}[n]$
System
$\mathrm{S}$

### Impulse Response

• Because of the superposition principle, for LTI systems the impulse response $\mathrm{h}[n]$ is sufficient to determine the response of the system to any input
• By using the unit impulse, any input sequence $\mathrm{x}[n]$ can also be written as
$\mathrm{x}[n] = \sum\limits_k \mathrm{x}[k] \, \mathrm{\delta}[n-k]$
• With $\mathrm{h}[n] = \mathrm{S}\left(\mathrm{\delta}[n])\right)$ and the superposition principle the following holds for the output signal $\mathrm{y}[n]$ of the system:
$\begin{array}{ll} \mathrm{y}[n] & = \mathrm{S}\left(\sum\limits_k \mathrm{x}[k] \, \mathrm{\delta}[n-k] \right) \\ & = \sum\limits_k \mathrm{x}[k] \, \mathrm{S}\left(\mathrm{\delta}[n-k] \right) \\ & = \sum\limits_k \mathrm{x}[k] \, \mathrm{h}[n-k] \\ & = \mathrm{x}[n] \ast \mathrm{h}[n] \\ \end{array}$
• This means, the response of an LTI system to any input can be determined by convolving the input with the system's impulse response

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