Suppose the inhabitants of my world wish to build a time machine capable of receiving information sent from the future.
It is only necessary to build the receiver; building the transmitter can be figured out in the future.
And it should be way easier to build the receiver.
What would be an intelligent design for such a receiver?
The sender(s) will solve your problems for you. The people in the future will already know how to do all this. They will know how to send and receive, and will send that information in the very first transmission.
So how do they send a message to a past that doesn't already have the tech? They downgrade their signal and send it back to a regular modern day device (cell phone, laptop, whatever) of a specific person, a character you create who will be the spearhead for the whole operation, someone who has already started work on the time machine(?)
How is it technically done? Doesn't matter. We don't know how time travel works, so it can be whatever you say (wormhole, faster than light speed signal, anything). Nobody asks how time travel works anymore. We just suspend our disbelief. How many Terminator movies have we seen without asking that question?
You're right, the receiver is the easy part. You want a system that will tolerate small inaccuracies in time and space since your transmitter may not be able to send a signal from the exact right place. In the future there might be a military base on the location of your receiver and the transmitter has to send from somewhere else. And you don't want to have to dedicate a huge chunk of time machine resources to the message sender. If you want to send a message in the most reliable and proven way, I suggest using ham radio frequencies to send Morse code.
You can learn a little more about it here.
Receiving considering time and space
The thing is that time and space are linked. Even without that link it's obvious there is a problem. The moment you travel back a day, you'll fly in outer space, very roughly that's 259000 km (1608000 miles) from the Earth (quite a lot of rounding errors possible, but you get the idea). Unfortunately our solar system is also moving with about 720000km/h (448000mph) and the whole galaxy is moving at 2.1million km/h (1.3 million mph). To further complicate the problem is that we don't know how fast we're exactly moving. We can only measure in relation to other objects. We could probably get a good idea due to the fixed speed of light, but as far as I can see we don't know our exact speed.
To get a signal towards us, we would either need to make something travel through time and space to the exact coordinates, which can then be easily received by even a short ranged bluetooth if you created that as a signal. Else you have a problem. You would need to timetravel the signal and then have it expand towards us. We don't know exactly where we're going, so it needs to expand to all sides. The problem here is signal strength. The further it is away, the less of the signal reaches Earth. It's like shining a torch towards a mountain kilometers away. Very little can be caught, regardless of the signal used. Only if you would know the exact location you can sent a directed beam, like a laser, for barely any loss.
So whatever you do, get something that is light speed, is powerful, doesn't degrade over distance and is unique. If you send out light it can be difficult to differentiate from all the other light. But if you sent out a radiowave that can penetrate the atmosphere with little change (long waved once for example) and is unique on Earth you have the best chance of picking it up.
To further augment this, have sattelites in orbit around the Earth, or even better, around the solar system. They have the least interference and can then reliably send the information to the Earth.
Buy this radio with three dollars, tune it to any frequency that records their broad casts and note your location, frequency, and time. Post this information somewhere on the internet. The future sender will send you a message at that frequency by playing a signal that destructively interferes with the signal you would normally receive, and then add the signal they want to send. Since the radio broadcast is recorded, knowing the signal to invert will be easy.
I would assume it isn't really important what your receiver is based on, radio waves, light or what ever. Since the future guys will know that and tweak there transmitter. Also less important should be the position, as long as it is well enough publicly known.
I would argue your receiver would need some other hardware, to log as much information as possible. So your receiver is not only receiving but maybe instead a time capsule it self? Like this the future people exactly know how your receiver will work. If you put in a logging device that keeps track of power usage, how much power is available, a gyro for orientation, they should have enough information to calculate when there are the highest chances to send something.
The aim would be to give them enough information so that even little events like, someone knocked against the table, a short blackout or anything like that isn't in the way of receiving.
Without knowing the rules of time travel in your world...
The unfortunate truth about time travel is that, insofar as we know, it can't happen. That means we're inventing an idea, but we're missing the rules of your world that are the foundation for the idea. (Normally we'd vote to close a question that doesn't have all the details, but this is your first question.)
One of the bigger problems is that @Trioxidane is correct, the Earth rotates and orbits. The solar system itself also orbits. That means that if you don't account for both spatial and temporal conditions, there's nothing on the receiving end but empty space. But if you think about it... that's boring (unless you're writing a story that explores that specific facet of time travel).
Another big problem is that it's "reasonably well understood" (*cough*) how to travel forward in time — you just travel as close as you can to the speed of light. When you get to your destination, time will have moved for them faster than for you. But backwards doesn't work that way — or in any way that we can "factually hypothesize."
So, without your world's rules for time travel we don't know how to resolve paradoxes or inconsistencies. For example, we could answer your question by suggesting the discovery of a tachyonic particle that would allow for communication across time. Your receiver is built to detect such signals — but then what? If it never detects a signal, did the future find that the discovery actually didn't work as advertised? Or does the future use a modulation that the past isn't demodulating? Or maybe the future simply doesn't have anything to say to the past? The fact that the solution worked using a fictional particle of matter is another form of black-boxing the solution or saying "magic happens here."
If we act without knowing any of your world's time travel rules, then perhaps the only sensible design is to internationally allocate an EM frequency that's reserved for time-traveling communication — and then hope that the unbelievers don't mess with the signal. After all, other than telling you what will happen in your near-future, how could you prove that a signal had traversed time? (And that's assuming the future doesn't have what Star Trek calls a "temporal prime directive," which means we're back to you not getting a signal).
And this assumes that the future hasn't found a way to manipulate communications in the past, meaning the "receiver" is nothing more than your cell phone.
Finally, let me point out that it might not even be in your best interest to look too closely at how to implement your receiver.1, 2 Such a receiver is, from a practical perspective, Clarkean magic. And it usually causes more problems to explain the magic than it does to simply accept it exists and move on with your story (in which case, the receiver is nothing more than the manifestation of your worldbuilding rule that communication across time is possible in your world). This idea is reflected in the movie "Frequency" where regular ham radios are used to communicate through time thanks to an undefined quirk of nature. (Or you could use the solution used in the movie "Primer," which is quite literally, "magic happens here." The time-traveling component of the story isn't explained at all... it just exists.)
Having said all that, we could provide better answers, but it would help to know the rules of time travel in your world. Thanks!
