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We visited the engineering and This book was developed based on our teaching of undergraduate- and graduate-level courses in digital signal processing over the past several years. In this book we present the fundamentals of discrete-time signals, systems, and modern digital processing as well as applications for students in electrical Springer, This book presents an introduction to the principles of the fast Fourier transform FFT.

It covers FFTs, frequency domain filtering, and applications to video and audio signal processing. As fields like communications, speech and image processing, and related areas are rapidly developing, the FFT as one of the essential parts in digital signal The crackle of random noise can be removed.

Maybe similar "sound recipes" can be compared music recognition services compare recipes, not the raw audio clips. If computer data can be represented with oscillating patterns, perhaps the least-important ones can be ignored. If a radio wave is our signal, we can use filters to listen to a particular channel.

In the smoothie world, imagine each person paid attention to a different ingredient: Adam looks for apples, Bob looks for bananas, and Charlie gets cauliflower sorry bud. The Fourier Transform is useful in engineering, sure, but it's a metaphor about finding the root causes behind an observed effect.

Labeling a circular path as a "complex sinusoid" is like describing a word as a "multi-letter". You zoomed into the wrong level of detail. Words are about concepts, not the letters they can be split into! The Fourier Transform is about circular paths not 1-d sinusoids and Euler's formula is a clever way to generate one:.

Must we use imaginary exponents to move in a circle? But it's convenient and compact. And sure, we can describe our path as coordinated motion in two dimensions real and imaginary , but don't forget the big picture: we're just moving in a circle. Let's say we're chatting on the phone and, like usual, I want us to draw the same circle simultaneously.

You promised! What should I say? I could say "2-inch radius, start at 45 degrees, 1 circle per second, go! We can even combine paths: imagine tiny motorcars, driving in circles at different speeds. The combined position of all the cycles is our signal, just like the combined flavor of all the ingredients is our smoothie. Based on this animation , here's the source code. Modern browser required. Now let's add a 2Hz cycle to the mix.

The little motorcars are getting wild: the green lines are the 1Hz and 2Hz cycles, and the blue line is the combined result. Try toggling the green checkbox to see the final result clearly. The combined "flavor" is a sway that starts at the max and dips low for the rest of the interval.

The yellow dots are when we actually measure the signal. In this case, cycles [0 1 1] generate the time values [2 -1 -1] , which starts at the max 2 and dips low We can't forget phase, the starting angle! Use magnitude:angle to set the phase. So [0 ] is a 1Hz cycle that starts at 45 degrees:.

This is a shifted version of [0 1]. On the time side we get [. The Fourier Transform finds the set of cycle speeds, amplitudes and phases to match any time signal. Our signal becomes an abstract notion that we consider as "observations in the time domain" or "ingredients in the frequency domain". Enough talk: try it out!

In the simulator, type any time or cycle pattern you'd like to see. If it's time points, you'll get a collection of cycles that combine into a "wave" that matches your desired points. But… doesn't the combined wave have strange values between the yellow time intervals? But who's to say whether a signal travels in straight lines, or curves, or zips into other dimensions when we aren't measuring it?

It behaves exactly as we need at the equally-spaced moments we asked for. Can we make a spike in time, like 4 0 0 0 , using cycles? I'll use parentheses for a sequence of time points, and brackets [] for a sequence of cycles. Although the spike seems boring to us time-dwellers one data point, that's it? Our cycle ingredients must start aligned at the max value, 4 and then "explode outwards", each cycle with partners that cancel it in the future.

Every remaining point is zero, which is a tricky balance with multiple cycles running around we can't just "turn them off". At time 0, the first instant, every cycle ingredient is at its max. Ignoring the other time points, 4? Imagine a constellation of points moving around the circle. Here's the position of each cycle at every instant:. When our cycle is 4 units long, cycle speeds a half-cycle apart 2 units will either be lined up difference of 0, 4, 8… or on opposite sides difference of 2, 6, 10….

When every cycle has equal power and 0 phase, we start aligned and cancel afterwards. I don't have a nice proof yet -- any takers? In my head, I label these signals as "time spikes": they have a value for a single instant, and are zero otherwise the fancy name is a delta function. Here's where phase comes in. Imagine a race with 4 runners.

Normal races have everyone lined up at the starting line, the 4 0 0 0 time pattern. What if we want everyone to finish at the same time? Just move people forward or backwards by the appropriate distance. Maybe granny can start 2 feet in front of the finish line, Usain Bolt can start m back, and they can cross the tape holding hands.

Phase shifts, the starting angle, are delays in the cycle universe. Here's how we adjust the starting position to delay every cycle 1 second:. If time points 4 0 0 0 are made from cycles [1 1 1 1] , then time points 0 4 0 0 are made from [1 ]. Note: I'm using "1Hz", but I mean "1 cycle over the entire time period". Test your intuition: Can you make 0 0 4 0 , i.

The big insight: our signal is just a bunch of time spikes! If we merge the recipes for each time spike, we should get the recipe for the full signal. This was my most challenging article yet.

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