Digital Signal Processing
Marrit Smit
June 2026
1 Week 1
Discrete signals can be obtained by continuous signals using sampling t = nT, where T is the sampling time. An
even signal is the same for negative time x(t) = x(−t), whereas odd will flip the value x(t) = −x(−t), where any
signal can be written as a combination of the two through xe,o (t) = x(t)±x(−t)
2 . A single frequency can be written
as x(t) = Aejω0 t = A(cos(ω0 t) + jsin(ω0 t), represented in the complex plane as an angle of ω0 t and a length of A.
This makes any frequency separated by 2π the same, as ej2πn = 1 for all n. This periodicity makes it such that
ω0 = 2πM
P . Exponential systems can be represented with x(t) = e
at
or x[n] = an .
A switch to the frequency domain holds under scalar multiplication and summation. A frequency shift is done
by multiplication with ej2πf0 t in the time domain such that the frequency is subtracted with f0 . Scaling in time
becomes inverse scaling in the time domain.
1.1 LTI Systems
If signals pass through a system without interacting, it is additive. For linear systems, the output scales with the
input - we can superimpose. Time invariant systems will give the same output for some input at whatever time.
Causal systems only depend on previous or current inputs, whereas the anticausal systems can also depend on future
inputs.
Memoryless systems only depend on the same timestep as their output, like with simultaneously causal and
anticausal systems. However, they need to be both of these. Stable systems are bounded for all time, meaning they
never go to infinity and have a constant upper limit. This is also called BIBO.
Convolving with a delta function remains the same function. Also, order of multiplication does not matter, such
that brackets can be computed out or in as well. Convolution in the time domain is multiplication in the frequency
domain. Shifting in time results in modulation in frequency. Reversal stays the same in both domains. Complex
conjugation in one domain results in the same, but reversed in the other domain. Correlation is convolution without
flipping the kernel, which can help detect repeating patterns.
Z-transforms can be one or two sided, dependant on how whether it’s 0 or infinity to infinity. Zeroes are when
the fraction goes to zero (top = 0) or poles when bottom = 0. DTFT is a special case of the z-transform. If the
radius of convergence is in the unit circle, |z| = 1. If it does not converge for |z| = 1. then the DTFT does not exist.
1.2 Tutorial 1
A signal is characterised by amplitude, phase and frequency or period: y(T ) = amp ∗ sin(freq ∗ t + phase). The
angular frequency is 2π/T = 2π ∗ f . We can describe systems by their impulse response. Every input signal can
be seen as a sum of infinite Dirac deltas, where any impulse response cam be computed by the convolution of the
input signal with the transfer function. This is simplified by switching to the frequency domain, where it becomes
multiplication.
If a system is LTI, the input signal can be seen as the sum of a sinusoid. To represent the transfer function, a
Bode plot is used, with a log scale
√ for frequency with the magnitude response in decibels: HdB = 20log10 (|H|). The
magnitude can be computed as ℑ2 + ℜ2 and the phase as arctan( ℑ ℜ ).
2 Week 2
2.1 DFT & FFT
DFT P is Discrete Fourier Transform, which transforms a sum in time domain to a sum in frequency domain through
N −1
X[k] = n=0 x[n]e−j2πkn/N . The DFT are samples of the DTFT (continuous result) X(ejω ) for each point ωk = 2πk N ,
which in the z-plane are uniformly spaced points on the unit circle. The inverse DFT is simply with a reciprocal of
N and a negative exponent. Positive about this, is that it’s computable since it’s a finite summation. The choice of
sampling frequency for a DFT can result in leakage, such that frequencies are picked up which are not the intended
one.
The DTFTP maps a discrete signal to a continuous transform, representing the spectrum. The DTFT of the dirac
impulse is 1 : δ[k]e−jΩk = 1, where the inverse of the DTFT is
1
, 1
R 2π
2π 0
X(ejω )ejωn dω.
The frequency response can always be written as the polar form: |H(ejω )|ejϕ(ω) , where the first part is the
magnitude and the second part is the phase. For real filters (where for all n, the values are real), the frequency response
satisfies Hermitian symmetry: its complex conjugate differs by only a minus sign before j: H(ejω ) = H ∗ (e−jω ).
This causes the magnitude to be an even function (since the absolute signs make it positive regardless of j), and the
phase odd, since ϕ(ω) = −ϕ(−ω).
2.2 Filters
Gaussian noise is noise with average zero and a certain variance σ 2 - the noise power. The summation of noise
will tend to be a Gaussian, which doesn’t change in f-domain. This noise can be used in modelling - Additive white
Gaussian noise (added to some model, broadband noise). Decibel conversion is xdb = 10log10 (x). To measure the
Psign 2
amount of noise: SNR: Pnoise = σσ2 or in decibels: Psign,dB − Pnoise,dB , where power in dB is: xdB = 20log10 .
Filters can be analogue (high/low/band pass/stop with RC) or digital (z-transforms). Passive filters do not require
power (RCL circuits), whereas active filters do (opamps). Digital filters can be made to form transfer functions using
multipliers (arrow with a), adders of signals (touching lines) and unit delay/storing signals (z −1 ).
These can be done parallel where the signal is split into multiple transfer functions, which can be added. Or, we
use a cascade: such that two different transfer functions are set to the same signal in sequence, which is the same as
computing its product. Or, we use a feedback loop, such that we get the transfer function H(z)/(1 − G(z)H(z)).
Types The moving average filter computes the average of the current and past input, which has the structure of a
1/2 multiplier, and a storage of memory. The impulse response would be two halves of 0 and 1. The first difference
operator uses half the difference between current and past input, making the 1 impulse response negative and adding
a negative multiplier in the structure. Average is low-pass and first difference is high pass.
Furthermore, we have Finite and infinite impulse response. The first is easy to design andP stabilise, but the
N −n
second is more efficient with CPU and memory. A FIR filter can be computed with H(z) = n=0 = h[n]z ,
where N is the order of the filter with length N+1. It contains a delay chain where each delay is multiplied with its
respective transfer function, and added together (discrete convolution).
One can also change filters by setting signs differently. A LP-HP change would be a minus sign in the storage
(z −1 → −z −1 ). LP to BS would be a square of the storage (z −2 ) and LP to BP is a minus square (−z −2 ).
Finally, we have a linear-phase digital filter, which changes the phase as ϕ(ω) = −Kω for some real K, which
helps in image processing.
2.3 Tutorial
If we knowPthe impulse response, we can compute any output through convolution:
∞
y[n] = x[k]h[n − k] for discrete time signals. We can do so manually by flipping the signal, then multiplying
with each position, which would result in some sum - the new value for that spot.
For causal systems, the ROC is everywhere outside the radius of a circle with the radius as the largest pole
magnitude. For anticausal systems, it is flipped - the ROC is always inside the circle made from the smallest pole
magnitude. Only iff the ROC includes |z| = 1, the Fourier transform exists.
3 Week 3
Recursive systems have outputs that depend on past outputs, where the order of the equation is how many outputs
are used. For instance, a first order only uses the previous output: y[n] = py[n − 1] + x[n]. The impulse response of
such a system canPbe computed by computing n = 0, n=1, etc. The transfer function H(z) can then be computed
through H(z) = h[n]z −n .
The system is causal and stable iff the pole is inside the unit circle of the z-plane. This means that it can be
between 0-1 (low pass) where the magnitude drops at higher frequencies and -1-0 (high pass).
To draw a system (recursive), one can use Pa delay element z −1 that includes the previous output (memory). For
−k
these systems, it is easier to find H(z) as: Pak z −k and then setting z = ejω to find the frequency response (only
1+ bk z
possible if ROC ∈ unit circle).
3.1 Tutorial
Psignal A
The Signal to noise ratio can be computed by the fraction of their respective power: SN R = Pnoise = 20log10 ( A ),
where the factor of 10 becomes 20 due to the power 2.
2
Marrit Smit
June 2026
1 Week 1
Discrete signals can be obtained by continuous signals using sampling t = nT, where T is the sampling time. An
even signal is the same for negative time x(t) = x(−t), whereas odd will flip the value x(t) = −x(−t), where any
signal can be written as a combination of the two through xe,o (t) = x(t)±x(−t)
2 . A single frequency can be written
as x(t) = Aejω0 t = A(cos(ω0 t) + jsin(ω0 t), represented in the complex plane as an angle of ω0 t and a length of A.
This makes any frequency separated by 2π the same, as ej2πn = 1 for all n. This periodicity makes it such that
ω0 = 2πM
P . Exponential systems can be represented with x(t) = e
at
or x[n] = an .
A switch to the frequency domain holds under scalar multiplication and summation. A frequency shift is done
by multiplication with ej2πf0 t in the time domain such that the frequency is subtracted with f0 . Scaling in time
becomes inverse scaling in the time domain.
1.1 LTI Systems
If signals pass through a system without interacting, it is additive. For linear systems, the output scales with the
input - we can superimpose. Time invariant systems will give the same output for some input at whatever time.
Causal systems only depend on previous or current inputs, whereas the anticausal systems can also depend on future
inputs.
Memoryless systems only depend on the same timestep as their output, like with simultaneously causal and
anticausal systems. However, they need to be both of these. Stable systems are bounded for all time, meaning they
never go to infinity and have a constant upper limit. This is also called BIBO.
Convolving with a delta function remains the same function. Also, order of multiplication does not matter, such
that brackets can be computed out or in as well. Convolution in the time domain is multiplication in the frequency
domain. Shifting in time results in modulation in frequency. Reversal stays the same in both domains. Complex
conjugation in one domain results in the same, but reversed in the other domain. Correlation is convolution without
flipping the kernel, which can help detect repeating patterns.
Z-transforms can be one or two sided, dependant on how whether it’s 0 or infinity to infinity. Zeroes are when
the fraction goes to zero (top = 0) or poles when bottom = 0. DTFT is a special case of the z-transform. If the
radius of convergence is in the unit circle, |z| = 1. If it does not converge for |z| = 1. then the DTFT does not exist.
1.2 Tutorial 1
A signal is characterised by amplitude, phase and frequency or period: y(T ) = amp ∗ sin(freq ∗ t + phase). The
angular frequency is 2π/T = 2π ∗ f . We can describe systems by their impulse response. Every input signal can
be seen as a sum of infinite Dirac deltas, where any impulse response cam be computed by the convolution of the
input signal with the transfer function. This is simplified by switching to the frequency domain, where it becomes
multiplication.
If a system is LTI, the input signal can be seen as the sum of a sinusoid. To represent the transfer function, a
Bode plot is used, with a log scale
√ for frequency with the magnitude response in decibels: HdB = 20log10 (|H|). The
magnitude can be computed as ℑ2 + ℜ2 and the phase as arctan( ℑ ℜ ).
2 Week 2
2.1 DFT & FFT
DFT P is Discrete Fourier Transform, which transforms a sum in time domain to a sum in frequency domain through
N −1
X[k] = n=0 x[n]e−j2πkn/N . The DFT are samples of the DTFT (continuous result) X(ejω ) for each point ωk = 2πk N ,
which in the z-plane are uniformly spaced points on the unit circle. The inverse DFT is simply with a reciprocal of
N and a negative exponent. Positive about this, is that it’s computable since it’s a finite summation. The choice of
sampling frequency for a DFT can result in leakage, such that frequencies are picked up which are not the intended
one.
The DTFTP maps a discrete signal to a continuous transform, representing the spectrum. The DTFT of the dirac
impulse is 1 : δ[k]e−jΩk = 1, where the inverse of the DTFT is
1
, 1
R 2π
2π 0
X(ejω )ejωn dω.
The frequency response can always be written as the polar form: |H(ejω )|ejϕ(ω) , where the first part is the
magnitude and the second part is the phase. For real filters (where for all n, the values are real), the frequency response
satisfies Hermitian symmetry: its complex conjugate differs by only a minus sign before j: H(ejω ) = H ∗ (e−jω ).
This causes the magnitude to be an even function (since the absolute signs make it positive regardless of j), and the
phase odd, since ϕ(ω) = −ϕ(−ω).
2.2 Filters
Gaussian noise is noise with average zero and a certain variance σ 2 - the noise power. The summation of noise
will tend to be a Gaussian, which doesn’t change in f-domain. This noise can be used in modelling - Additive white
Gaussian noise (added to some model, broadband noise). Decibel conversion is xdb = 10log10 (x). To measure the
Psign 2
amount of noise: SNR: Pnoise = σσ2 or in decibels: Psign,dB − Pnoise,dB , where power in dB is: xdB = 20log10 .
Filters can be analogue (high/low/band pass/stop with RC) or digital (z-transforms). Passive filters do not require
power (RCL circuits), whereas active filters do (opamps). Digital filters can be made to form transfer functions using
multipliers (arrow with a), adders of signals (touching lines) and unit delay/storing signals (z −1 ).
These can be done parallel where the signal is split into multiple transfer functions, which can be added. Or, we
use a cascade: such that two different transfer functions are set to the same signal in sequence, which is the same as
computing its product. Or, we use a feedback loop, such that we get the transfer function H(z)/(1 − G(z)H(z)).
Types The moving average filter computes the average of the current and past input, which has the structure of a
1/2 multiplier, and a storage of memory. The impulse response would be two halves of 0 and 1. The first difference
operator uses half the difference between current and past input, making the 1 impulse response negative and adding
a negative multiplier in the structure. Average is low-pass and first difference is high pass.
Furthermore, we have Finite and infinite impulse response. The first is easy to design andP stabilise, but the
N −n
second is more efficient with CPU and memory. A FIR filter can be computed with H(z) = n=0 = h[n]z ,
where N is the order of the filter with length N+1. It contains a delay chain where each delay is multiplied with its
respective transfer function, and added together (discrete convolution).
One can also change filters by setting signs differently. A LP-HP change would be a minus sign in the storage
(z −1 → −z −1 ). LP to BS would be a square of the storage (z −2 ) and LP to BP is a minus square (−z −2 ).
Finally, we have a linear-phase digital filter, which changes the phase as ϕ(ω) = −Kω for some real K, which
helps in image processing.
2.3 Tutorial
If we knowPthe impulse response, we can compute any output through convolution:
∞
y[n] = x[k]h[n − k] for discrete time signals. We can do so manually by flipping the signal, then multiplying
with each position, which would result in some sum - the new value for that spot.
For causal systems, the ROC is everywhere outside the radius of a circle with the radius as the largest pole
magnitude. For anticausal systems, it is flipped - the ROC is always inside the circle made from the smallest pole
magnitude. Only iff the ROC includes |z| = 1, the Fourier transform exists.
3 Week 3
Recursive systems have outputs that depend on past outputs, where the order of the equation is how many outputs
are used. For instance, a first order only uses the previous output: y[n] = py[n − 1] + x[n]. The impulse response of
such a system canPbe computed by computing n = 0, n=1, etc. The transfer function H(z) can then be computed
through H(z) = h[n]z −n .
The system is causal and stable iff the pole is inside the unit circle of the z-plane. This means that it can be
between 0-1 (low pass) where the magnitude drops at higher frequencies and -1-0 (high pass).
To draw a system (recursive), one can use Pa delay element z −1 that includes the previous output (memory). For
−k
these systems, it is easier to find H(z) as: Pak z −k and then setting z = ejω to find the frequency response (only
1+ bk z
possible if ROC ∈ unit circle).
3.1 Tutorial
Psignal A
The Signal to noise ratio can be computed by the fraction of their respective power: SN R = Pnoise = 20log10 ( A ),
where the factor of 10 becomes 20 due to the power 2.
2
