! Discrete Time Signals. ! Signal Properties. ! Discrete Time Systems. ! Signals carry information. ! Examples:

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Kristina Beasley
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1 Lecture Outline ESE 531: Digital Signal Processing Lec 2: January 17, 2017 Discrete Time Signals and Systems! Discrete Time Signals! Signal Properties! Discrete Time Systems 2 Discrete Time Signals Signals! Signals carry information! Examples: " Speech signals transmit language via acoustic waves " Radar signals transmit the position and velocity of targets via electromagnetic waves " Electrophysiology signals transmit information about processes inside the body " Financial signals transmit information about events in the economy! Signal processing systems manipulate the information carried by signals 3 4 Signals are Functions A Menagerie of Signals 5 6 1

2 Plotting Signals Correctly Unit Sample 7 8 Unit Step Unit Pulse 9 10 Real Exponential Sinusoids

3 Sinusoid Examples Sinusoid in Matlab Complex Sinusoid Complex Sinusoid as Helix Animation: Negative Frequency Phase of a Sinusoid

4 Complex Exponentials Complex Exponentials Bounded Unbounded Digital Signals Signal Properties Finite/Infinite Length Sequences Windowing

5 Zero Padding Periodic Signals Periodization Causal Signals Even Signals Odd Signals

6 Signal Decomposition Decomposition Example Decomposition Example Decomposition Example Decomposition Example Discrete-Time Sinusoids

Property #1: Aliasing of Sinusoids Aliasing Example 37 38 Aliasing Example Alias-Free

7 Property #1: Aliasing of Sinusoids Aliasing Example Aliasing Example Alias-Free Frequencies Which is higher in frequency? Low and High Frequencies! cos(πn) or cos(3π/2n)?

Increasing Frequency Decreasing Frequency 43 44

of Sinusoids 45 46 Harmonic Sinusoids Periodic or

8 Increasing Frequency Decreasing Frequency Property #2: Periodicity of Sinusoids Aperiodicity of Sinusoids Harmonic Sinusoids Periodic or not?! cos(5/7πn)! cos(π/5n)! What are N and k?

9 Periodic or not?! cos(5/7πn) " N=14, k=5 " cos(5/14*2πn) " Repeats every N=14 samples! cos(π/5n) " N=10, k=1 " cos(1/10*2πn) " Repeats every N=10 samples Periodic or not?! cos(5/7πn) " N=14, k=5 " cos(5/14*2πn) " Repeats every N=14 samples! cos(π/5n) " N=10, k=1 " cos(1/10*2πn) " Repeats every N=10 samples! cos(5/7πn)+cos(π/5n)? Periodic or not?! cos(5/7πn)+cos(π/5n)? " N=SCM{10,14}=70 Discrete-Time Systems " cos(5/7*πn)+cos(π/5n) " n=n=70#cos(5/7*70π)+cos(π/5*70)=cos(25*2π)+cos(7*2π) 51 Discrete Time Systems Signal Length and Systems

10 System Examples System Examples System Properties Memoryless! Memoryless! Linearity! Time Invariance! y[n] depends only on x[n]! Causality! BIBO Stability! Examples:! Ideal delay system (or shift system): " y[n]=x[n-m] memoryless?! Square system: " y[n]=(x[n]) 2 memoryless? Linear Systems Proving Linearity

11 Linearity Example: Moving Average Linearity Example: Moving Average Example: Squaring is Nonlinear Time-Invariant Systems Example: Moving Average Example: Decimation

12 Causal Systems Stability! BIBO Stability " Bounded-input bounded-output Stability! Forward difference system: " y[n]=x[n+1]-x[n] causal?! Backward difference system: " y[n]=x[n]-x[n-1] causal? Examples Big Ideas! Causal? Linear? Time-invariant? Memoryless? BIBO Stable?! Time Shift: " x[n] y[n] = x[n-m] = m]! Accumulator: " y[n] y[n] = k= x[k]! Compressor (M>1): y[n] = x[mn] n! Discrete Time Signals " Unit impulse, unit step, exponential, sinusoids, complex sinusoids " Can be finite length, infinite length " Properties " Even, odd, causal " Periodicity and aliasing " Discrete frequency bounded!! Discrete Time Systems " Transform one signal to another " Properties " Linear, Time-invariance, memoryless, causality, BIBO stability Admin! Enroll in Piazza site: " piazza.com/upenn/spring2017/ese531! HW 1 out Thursday 71 12

ESE 531: Digital Signal Processing

ESE 531: Digital Signal Processing Lec 10: February 14th, 2017 Practical and Non-integer Sampling, Multirate Sampling Lecture Outline! Downsampling/Upsampling! Practical Interpolation! Non-integer Resampling!