Compiled DSP Questions: Answers 26–50 (Signal Operations, Systems, Z-Transform, FFT)

Worked answers to questions 26-50 of the compiled CSE-409 DSP question list, including convolution, system classification, inverse Z-transform and block diagrams.

Posted Mar 19, 2026

By Sadman Ishtiak

views 10 min read

How This Post Fits the Series

Questions 26–50 below are copied in order from the Compiled DSP Questions for CSE-409 page. The answers are worked, and the “Read more” links go to the full treatment in this series.

Q26. Explain “signal” in a way that satisfies a teacher

A signal is a physical quantity that carries information and varies as a function of one or more independent variables, usually time. Mathematically it is a function $x(t)$ or a sequence $x[n]$. The word matters because DSP is the science of extracting, modifying, or transmitting that information. A voltage from a microphone, the pixels of an image, and the readings of a temperature sensor are all signals because each is a measurable quantity whose variation encodes something useful.

Q27. Is a digital photo of Einstein a signal? Explain

Yes. A photograph is a two-dimensional spatial signal. The scene’s brightness is sampled on a grid of pixels (sampling in space) and each brightness is stored as a finite set of integer levels (quantization). The result is a 2-D discrete signal $f[m,n]$ that can be filtered, compressed, enhanced, or analysed exactly like a 1-D signal. Image processing is therefore a direct branch of DSP.

A digital signal is a sequence of numbers in which both axes are discrete: time is sampled at fixed intervals and amplitude is quantized to a finite set of levels. It is represented by $x[n]$, where $n$ is an integer index and each $x[n]$ is a code word of $B$ bits. Because it is discrete in both dimensions, it can be stored, transmitted without further quality loss, and processed by a computer.

Q29. Is biometrics an application of DSP? List other applications

Yes. Fingerprint, face, iris, voice, and ECG biometrics all start with a sensor signal that must be enhanced, segmented, and converted into features; that is DSP. Matching and classification come after the DSP front end.

Other applications: speech coding and recognition, audio effects and noise cancellation, radar and sonar, medical imaging, digital communication (equalization, channel coding), seismology, vibration monitoring, and brain-computer interfaces.

Q30. Properties of auto-correlation and cross-correlation

Auto-correlation $R_{xx}[k]=\sum_n x[n]x[n+k]$:

$R_{xx}[0]=\sum_n x^2[n]$ is the energy (or power for power signals).
It is even: $R_{xx}[-k]=R_{xx}[k]$.
It is maximum at lag zero: $R_{xx}[0]\ge \vert R_{xx}[k]\vert$ for all $k$.
A periodic signal has a periodic auto-correlation with the same period.

Cross-correlation $R_{xy}[k]=\sum_n x[n]y[n+k]$:

It is not even in general.
$R_{xy}[k]=R_{yx}[-k]$.
It is bounded by $\vert R_{xy}[k]\vert\le\sqrt{R_{xx}[0]R_{yy}[0]}$.
The lag of its peak measures the time delay between the two signals.

Q31. System responses for $x[n]=\vert n\vert$, $-3\le n\le 3$

The input is ${3,2,1,0,1,2,3}$ at $n=-3,\dots,3$.

Identity: $y[n]=x[n]$, same sequence.
Delay by 1: $y[n]=x[n-1]$, the sequence moves one sample right.
Advance by 1: $y[n]=x[n+1]$, the sequence moves one sample left.
3-point moving average: $y[n]=\tfrac13(x[n-1]+x[n]+x[n+1])$. With zero-padding at the edges:

$n$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y[n]$	$5/3$	$2$	$1$	$2/3$	$1$	$2$	$5/3$

The middle values follow $\tfrac13(\vert n-1\vert+\vert n\vert+\vert n+1\vert)$; the edges are smaller because the missing neighbour is zero.

Q32. Convolution of $h[n]={1,2,1,-1}$ and $x[n]={1,2,3,1}$

Use $y[n]=\sum_k x[k]h[n-k]$. The result has length $4+4-1=7$:

\[y=\{1,\ 4,\ 8,\ 8,\ 3,\ -2,\ -1\}\]

Step by step:

$y[0]=1\cdot1=1$
$y[1]=1\cdot2+2\cdot1=4$
$y[2]=1\cdot3+2\cdot2+1\cdot1=8$
$y[3]=1\cdot1+2\cdot3+1\cdot2+(-1)\cdot1=8$
$y[4]=2\cdot1+1\cdot3+(-1)\cdot2=3$
$y[5]=1\cdot1+(-1)\cdot3=-2$
$y[6]=-1\cdot1=-1$

Q33. What is a discrete-time system?

A discrete-time system is an operator $T{\cdot}$ that maps an input sequence $x[n]$ to an output sequence $y[n]=T{x[n]}$. It is described by a difference equation, an impulse response $h[n]$, a block diagram, or a transfer function $H(z)$. Systems are classified as linear or nonlinear, time-invariant or time-varying, causal or non-causal, stable or unstable, and static or dynamic.

Q34. Block diagram for $y[n]=0.25y[n-1]+0.5x[n]+0.5x[n-1]$

The structure needs one delay for the input and one for the output, two adders, and three multipliers.

x[n] --+-->[0.5]--+--------------->(+)---> y[n]
       |          |                 ^
       +->[z^-1]--+-->[0.5]--+      |
                             |      |
y[n] --+---------------------+------+
       |
       +->[z^-1]-->[0.25]----+

In words: the input branch forms $0.5x[n]+0.5x[n-1]$; the feedback branch forms $0.25y[n-1]$; these two sums are added to produce $y[n]$. This is a first-order IIR filter with one pole at $z=0.25$.

Q35. Static vs dynamic, time-invariant vs time-variant, linear vs nonlinear

Pair	First kind	Second kind
Static vs dynamic	output depends only on the present input, e.g. $y[n]=2x[n]$	output depends on past or future inputs, e.g. $y[n]=y[n-1]+x[n]$
Time-invariant vs time-variant	a shift of the input shifts the output by the same amount	the rule itself changes with time, e.g. $y[n]=nx[n]$
Linear vs nonlinear	superposition holds: $T{ax_1+bx_2}=aT{x_1}+bT{x_2}$	superposition fails, e.g. $y[n]=x^2[n]$

Q36. Define quantization error

Quantization error is the difference between the quantized value and the true sample:

\[e[n]=Q(x[n])-x[n]\]

For a uniform quantizer with step $\Delta$ and rounding, $\vert e[n]\vert\le\Delta/2$. It is modelled as zero-mean uniform noise with power $\Delta^2/12$.

Q37. Role of the number of bits $B$ in a quantizer

The number of bits sets the number of levels $L=2^B$ and therefore the step size $\Delta=R/2^B$ for a full-scale range $R$. Each extra bit halves $\Delta$, reducing quantization noise power by a factor of four, i.e. about $6.02$ dB better SQNR. More bits mean better fidelity but larger memory, more bandwidth, and more processing per sample. For a full-scale sinusoid,

\[\text{SQNR}\approx6.02B+1.76\ \text{dB}\]

Q38. How to convert an analog signal to digital

Transduce and condition the physical quantity into a voltage.
Anti-alias filter it to below $F_s/2$.
Sample at $F_s\ge2F_{\max}$ to get $x[n]$.
Quantize each sample to $2^B$ levels.
Encode each level as a binary word.

The result is a digital signal ready for storage or processing.

Q39. Applications of the Z-transform and definition of ROC

Applications: solving linear constant-coefficient difference equations, finding the system function $H(z)$, testing stability from pole locations, converting convolution into multiplication, analysing transient and steady-state response, and designing filters.

The region of convergence is the set of complex $z$ for which

\[X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n}\]

converges absolutely. It is an annulus centred at the origin, contains no poles, and is determined by the signal’s support (right-sided, left-sided, or two-sided).

Q40. Z-transform and ROC of ${1,0,\mathbf{3},-1,2}$

With the bold $3$ at $n=0$ and the sequence running $n=-2,\dots,2$:

\[X(z)=z^{2}+3-z^{-1}+2z^{-2}\]

Because the sequence is finite and includes samples on both sides of the origin, the ROC is the whole $z$-plane except $z=0$ and $z=\infty$.

If instead the sequence is taken as causal with $n=0,\dots,4$, then

\[X(z)=1+3z^{-2}-z^{-3}+2z^{-4},\qquad \vert z\vert>0\]

Q41. Inverse Z-transform of $X(z)=\dfrac{z+0.2}{(z+0.5)(z-1)}$, $\vert z\vert>1$

The ROC is outside the outermost pole, so the sequence is right-sided (causal). Expand $X(z)/z$:

\[\frac{X(z)}{z}=\frac{z+0.2}{z(z+0.5)(z-1)}\]

Residues:

at $z=0$: $\dfrac{0.2}{(0.5)(-1)}=-0.4$
at $z=-0.5$: $\dfrac{-0.3}{(-0.5)(-1.5)}=-0.4$
at $z=1$: $\dfrac{1.2}{(1.5)(1)}=0.8$

\[X(z)=-0.4-\frac{0.4z}{z+0.5}+\frac{0.8z}{z-1}\]

and therefore

\[x[n]=-0.4\delta[n]-0.4(-0.5)^n u[n]+0.8u[n]\]

Check: $x[0]=-0.4-0.4+0.8=0$, $x[1]=0.2+0.8=1$, and $x[\infty]=0.8$.

Q42. The FFT speed increase factor of 21.33

The direct DFT needs $N^2$ complex multiplications. The radix-2 FFT needs $\frac{N}{2}\log_2 N$. Their ratio is

\[\frac{N^2}{(N/2)\log_2 N}=\frac{2N}{\log_2 N}\]

For $N=64$:

\[\frac{2(64)}{6}=\frac{128}{6}\approx21.33\]

So the FFT is about $21.33$ times faster for a 64-point transform. The factor grows with $N$, which is why FFT made real-time spectral processing practical.

Q43. How the decimation-in-time FFT works

Split $x[n]$ into even-indexed and odd-indexed subsequences.
Each half-length sequence is transformed recursively.
Combine the two $N/2$-point results with butterflies:

\[X[k]=E[k]+W_N^k O[k],\qquad X[k+N/2]=E[k]-W_N^k O[k]\]

Repeat until only 2-point transforms remain, giving $\log_2 N$ stages.
The input must be in bit-reversed order; the output ends in natural order.

Each butterfly uses one complex multiplication and two additions, so the total cost is $(N/2)\log_2 N$ multiplications.

Read more: Radix-2 FFT and DIT vs DIF FFT.

Q44. DFT of ${1,1,0,0}$ and IDFT of ${1,0,1,0}$

DFT with $W_4=-j$: $X[k]=1+W_4^k$.

$X[0]=2$
$X[1]=1-j$
$X[2]=1-1=0$
$X[3]=1+j$

So $X={2,1-j,0,1+j}$.

IDFT of $X={1,0,1,0}$:

\[x[n]=\frac14\sum_{k=0}^{3}X[k]W_4^{-nk}=\frac14\left(1+(-1)^n\right)\]

giving $x={0.5,0,0.5,0}$.

Q45. Delay and advance on the time axis

$x[n-k]$ with $k>0$ is a delay: every sample occurs $k$ steps later, so the waveform moves right. $x[n+k]$ is an advance: the waveform moves left. Delay is physically realizable with memory; advance is non-causal and needs future samples.

For $x[n]={2,1,\mathbf{0},-1,-2}$: $x[2n]={2,0,-2}$ at $n=-1,0,1$ and $x[n/2]={2,0,1,0,0,0,-1,0,-2}$ for $n=-4,\dots,4$ (see Q16).

Q46. Role of variance in signal study

Variance measures the spread of a signal around its mean, i.e. the power of its fluctuating (AC) part. It quantifies uncertainty, noise level, and signal activity. It appears directly in SNR, in estimator quality (low variance means a reliable estimate), in feature extraction for classification, and in the design of detectors and adaptive filters.

Q47. Define and calculate skewness and kurtosis

For a data set with mean $\mu$ and standard deviation $\sigma$:

\[\text{skewness}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{x_i-\mu}{\sigma}\right)^3\] \[\text{kurtosis}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{x_i-\mu}{\sigma}\right)^4\]

Skewness measures asymmetry: zero is symmetric, positive means a longer right tail, negative a longer left tail. Kurtosis measures tail heaviness and peakedness; the normal distribution has kurtosis $3$, so excess kurtosis is kurtosis minus $3$.

For ${1,2,3,4,9}$: mean $3.8$, $\sigma\approx2.786$, skewness $\approx1.04$ (right-skewed), excess kurtosis $\approx-0.33$.

Q48. Define impulse

In discrete time the unit impulse is

\[\delta[n]=\begin{cases}1,&n=0\\0,&n\ne0\end{cases}\]

It is the identity element of convolution: $x[n]*\delta[n]=x[n]$, and any sequence can be written as $x[n]=\sum_k x[k]\delta[n-k]$.

In continuous time the Dirac delta $\delta(t)$ is a unit-area spike with the sifting property $\int x(t)\delta(t-t_0)\,dt=x(t_0)$.

Q49. Advantages of FIR over IIR filters

Can have exactly linear phase when coefficients are symmetric, so no phase distortion.
Always stable, because there is no feedback.
Finite impulse response means finite memory and no limit cycles.
Less sensitive to coefficient quantization.
Easy to design by windows, frequency sampling, or Parks-McClellan.
Simple to implement in hardware and to pipeline.

The price is a higher order and more computation for a sharp magnitude response.

Q50. Direct Form I for $y[n]=0.5y[n-1]-0.76y[n-2]+0.63y[n-3]+x[n]+0.875x[n-1]$

Rearrange into the transfer function:

\[H(z)=\frac{1+0.875z^{-1}}{1-0.5z^{-1}+0.76z^{-2}-0.63z^{-3}}\]

Direct Form I uses a chain of delays on the input and a separate chain on the output:

Input branch coefficients: $b_0=1$, $b_1=0.875$, and zeros for $b_2,b_3$.
Feedback branch coefficients: $a_1=-0.5$, $a_2=0.76$, $a_3=-0.63$ (the signs are already in the rearranged equation).

In block-diagram form: $x[n]$ feeds three series delays; the taps $1$ and $0.875$ are summed into the central adder. $y[n]$ feeds three series delays; the taps $0.5$, $-0.76$, $0.63$ are summed into the same adder. The adder output is $y[n]$. Three delays in each chain make six delays total for Direct Form I; Direct Form II would need only three.

Academic, CSE-409 Digital Signal Processing

This post is licensed under CC BY 4.0 by the author.