1. Lectures 5. Basics of Digital Signal Processing
Micro-Nanoscience
and Nanotechnology
Lectures 5. Basics of Digital Signal Processing
© 2013 Universitat Politècnica de Catalunya (EPSEVG)
José Antonio Soria Pérez - Associate Professor - ETSEIAT (UPC).
Office hours: EET - D126 (TR2, 2nd floor): TU 10-14h and 17-19h THU: 10-14h

2. Digital Systems: The starting point
Basics:
Analog signals contain “real world” information: (voice, image, environment data, ...)
Analyzing such information requires of the implementation of processing systems (either LTI or its equivalent digital version).
Amplifiers and Filters (either analog or digital) are considered the most basic form of signal processing
The implementation of electronic digital systems requires of two steps:
Passing from Analog to a Digital domain (use of AD Converters)
Implement the digital operation at hand (use of FPGAs, μC, μP or DSP)
Real World
Sensor
Analog Signal
Conditioning
Digital
Processor
ADC
Filter
Output

3. Signals: Continuos vs Discrete
Analog domain
In the analog domain, x is well defined for any time t within the range of interest
Digital domain
In the digital domain, x is only valid for n integer samples within the time interval t.
Undefined value within two valid samples . T is known as the sampling rate x t x n

4. Discrete sequences. Representations
Graphical
Function
x(n) 5
, per -3 ≤ n ≤ 0 4 3
, per 1 ≤ n ≤ 3 2
, per qualsevol altre cas 1
Tabulated
Vector n
x(n) = {... -4, 0, 0, 0, 0, 1, 1, 1, 4, 5 }
x(n)

5. Discrete Signals. Examples
Exponential: x(n) = an, for all n
Impulse: δ(n)
δ(n)
0 < a < 1
, for n = 0
, for n ≠ 0 n
a >1
Step: u(n) n
u(n)
, for n ≥ 0
, for n < 0
-1 < a < 0 n
Ramp: ur(n)
-1 < a
ur(n)
, for n ≥ 0 n
, for n < 0

6. Discrete sinusoids Main properties:
Only periodic if f is a division of two integers f = k/N. This means there must be some k such that f ≤ ½, for N being the total number of samples of x(n)
All frequencies separated by integer values of 2π are identical (The different ones are just those within -½ ≤ f ≤ ½ ( or -π ≤ Ω ≤ π). Frequencies out of this range are aliases from these.
Highest oscillating profile attainable only for f = ½ (and f = -½ ), or equivalently, Ω = π (Ω = -π).

7. Analog-to-Digital conversion
In practice, the source of signal information is continuous.
Information must be transformed to a sequence of numeric values in order to process it by means of digital devices
ADC (Analog-to-Digital Converter): Develops a three-step procedure: Sampling, Quantification and Coding
Sample& Hold
x(n)
xq (n)
x = {x1, x2, ..., xn }
xa(t)
Sampling
Quantification
Coding
Output vector
ADC
Step
approximation
Original
Signal
Hold
Magnitude
Sample t 2T 4T 6T 8T

8. Sampling Rate. Considerations
Generally, the ADC develops a uniform periodical sampling.
The sampling rate establishes a relation between the continuous and discrete variables by t = nT = n/Fs, which have severe implications in the frequency components of periodical signals
x(n)
x(n) = xa(nT)
Continuous-time
Signal
Discrete-time
Signal
Fs = 1/T
Sampler n
Fs.- Sampling rate
T 2T T T t = nT
Lost of Signal Information
(Lossy signals)

9. How to choose Fs? Sampling theorem
Given an analog signal xa(t) wih Fmax = B being its component of highest frequency, then, recovering xa(t) from its digital version x(nTs) is possible by means of the interpolating function
if, and only if, Fs > 2·Fmax ≡ 2·B
Nyquist rate: FS ≡FN=2B = 2Fmax
This theoreme is only useful from a theoretical pespective (N=∞). In practice, FS ≡ 10FN (rule-of-thumb) and linear interpolation is used in the estimation of xa(t)*.

10. Quantification error Estimation: Example: Fs = 1 Hz
In practice, the precission of values is limited to the number-of-bits of the ADC.
Estimation:
If Q stands for the process being used in the quantification of x(n), {xq(n)=Q[x(n)]: truncation or round-off), the error is denoted as eq(n)=xq(n)- x(n)
Example:
Fs = 1 Hz
1 significat digit (one decimal pos.)
11 quantification levels (L =11)
xa(t)
Quantification
level
xq(n)
, per n ≥ 0
1.0
0.9
, per n < 0
0.8
0.7 Δ
0.6
Quantf.
Range.
0.5
0.4
Level
Step
0.3
0.2
0.1 n
Maximum error is half the
level step of quantification level

11. SQNR
Signal Quantification Noise Ratio – Noise level generated during quantification in relation to the original signal
Sinus example
Signal power (error and input):
eq(t)= (Δ/2τ)t, τ is the time interval of xa(t) within quantification range, and b corresponds to the number of bits from the ADC
Quantified sample
xq(nT)
Not quantified sample
xa(nT)
ADC output
xq(nT)
Original Signal
xa(t) Δ
Δ/2 4Δ 3Δ 2Δ Δ t Δ
Magnitude 2A
-τ τ -Δ
eq(t)
-2Δ
Δ/2 -τ
-3Δ t
-4Δ τ
-Δ/2 n
T T 3T 4T T 6T T 8T T
Time

12. More about discrete-time signals
Concepts
Energy and power
Periodicity
Simetry
Basic transformations
Time shift
Reflection
Compression-Decompression
Arithmetic operations: scale, addition and pointwise multiplication

13. Energy vs Power Signal energy Considerations:
N .- Number of samples of x(n)
Considerations:
For finite E (0 < E < ∞) and P = 0, x(n) is an energy signal. This corresponds to all transition signals of zero final value x(∞)=0
P can be either finite or infinite, for E being finite. If P is finite, then x(n) is a power signal
Step, exponentials and periodical signals are examples of power signals. The ramp response is not La rampa no és cap de les dues
Signal power

14. Energia vs Potència Energia Consideracions: Potència
N .- Nombre de mostres de x(n)
Consideracions:
Si E és finit (0 < E < ∞), P = 0 i x(n) és un senyal d’energia. Aquest és el cas de tots els senyals que tenen un caràcter transitori
Si E és infinit, P pot ser tant finit com infinit. En el cas que P tingui un valor finit x(n) és un senyal de potència
Graons i exponencials i senyals periòdiques són exemples de senyals de potència. The ramp is none of these
Potència

15. Periodicity
A signal x(n) is said to be N-periodical (N > 0) if, and only if:
otherwise, it is aperiodical
x(n)
x(n)
Periodical
Aperiodical

16. Basic Transformations
x(n) -5
Time shift
Refletion
x(n-3)
x(-n) -2 5
Delay by 3
Basic
x(n+2)
x(-n+2) -7 6
Advance by 2
Composition

17. Simmetry A signal x(n) is symmetrical (or even) if x(n)= x(-n)
A signal x(n) is antisymètrical (or odd) if x(-n)= -x(n)
Any signal can be expressed as a composition of two components (even and odd)
x(n)
x(n)
Even
Odd
Even
Component
Odd
Component

18. Other transformations
x(n) -5
Downsampling (Compression)
Other
Scale: y(n)=A·x(n)
Addition: y(n)= x1(n)+ x2(n)
Pointwise multiplication: y(n)= x1(n)·x2(n)
Upsampling (Descompression)
y(n)=x(2n)
-10